Speaking as a VC who interviews for our own entry-level roles, I don't think your current pitch will necessarily hurt you, but it won't really help you stand out either for these reasons:

Not a unique pitch: It's actually the opposite of non-conventional to offer to work for free because the bar for making this offer is low. Many people can and do offer to work hard for free because you don't need experience or any other value add to be able to offer this.

Adverse signalling: You're sending the negative signal that you don't particularly value your own expertise, experience or network, since you're only pitching your willingness to work under relatively unfavorable terms and are not mentioning other qualities that would be far more pertinent to a VC job. For most VCs that are not on their first fund, an analyst salary is not a huge amount of money, and they are likely much more concerned about the potential lost productivity/invested training hours from hiring the wrong person, as opposed to paying the right person. Same idea for title - a title is cheap and easy to give out, as evidenced by the large # of non-GP "Partners" in the Valley, so saying you're willing to take any title doesn't do them a particularly big favor.

Junior VCs must find ways to create value, not just take direction: VCs don't enter the industry to manage employees, and the GPs are usually too busy with their board memberships and deal pipelines to spend much time actually specifying discrete workflows for their junior professionals. When you say that you're willing to work for free, but don't describe any creative, differentiated ways in which you can actually help the firm make more money, you're really creating the impression (at least for me) that it would require work to manage and train you. It takes time to devise tasks, communicate them clearly and then provide feedback afterward, and if they believe that you will require a large upfront time investment before you start becoming productive, then they'll be wary of hiring you. Some of the best people I've interviewed have shown up at our initial meeting equipped with actionable ideas and plans for how the firm can do better, whether that's on the deal sourcing, marketing, portfolio management or reporting fronts. These people give off the impression that on day one they would have the self-direction and internal motivation to drive their own projects through without needing much input from me, and that's an attractive proposition.

Some firms do hire from their intern pool, especially if you're an MBA summer associate, so if you can get into those recruiting processes, it's a solid and well traveled path into the industry. However, even in those cases, you won't win the full-time job offer on the basis of the pitch you're making in your question.

Below, I've written out some ideas for standing out, especially if you come from a non-traditional background (i.e. you were not previously a semi-successful founder, product manager or investment banker). Some people would advise you to earn work experience at a startup first, which is fine advice, but since it's also possible to go straight to a VC with some effort on your part, I'm going to focus on answering your question which I assume focuses on getting into VC today:

1) Demonstrate market knowledge with research decks around specific focus areas. Since you do not have a traditional background for investing, you'll need to demonstrate that you can learn enough on your own to identify interesting startups, develop a thesis and argue convincingly for certain investments. I would recommend putting together some simple presentations (or detailed emails) that outline investment recommendations in a few categories within healthcare (examples could be medical devices, EMR software, hospital administration SaaS, fitness trackers, health insurance shopping). Topics to cover include market size, competitive landscape, what startups have been VC-invested and by whom, business model and any other useful data that are not easily obtainable. One of my prior interviewees, when I asked him to look into a company for me, actually called 20 of their customers and asked to speak to the VP-level end buyer of the product to get direct feedback as well as estimates of contract sizes. That was pretty gutsy, outside of what I would have expected him to do on that time frame and of course I loved it. Once you have these mini-presentations, you can send them to any VC as a part of a cold email or follow-up conversation to convey what type of research and insight you'll be able to deliver immediately. This demonstrates both knowledge and proactiveness, and the document will be super easy to forward onto other partners, which helps you gain "virality" in an organization.

2) Start sourcing today. You don't need to get a job offer to get "a shot at helping source" deals, since sourcing just means bringing in possible investments with a discerning filter. You can get to know a few interesting healthcare startups now via meetups, conferences or your college network, and offer to introduce them to VCs you're talking to. No VC will be annoyed by this unless you're not filtering the companies at all, and you can stay top of mind for them over the course of several months by consistently pinging them with attractive opportunities, especially if you are able to make a persuasive argument for why those companies are worth a look. Even if the firm is not hiring immediately, you'll be a familiar name to them if they start looking.

3) Benefit their portfolio today. In a similar vein to point #2, you can start thinking up strategies for helping the VC's portfolio companies, potentially with hiring, marketing and business development opportunities. As a young person, you probably have a network of ex-classmates, at least some of whom are now software engineers, designers, marketers or salespeople. Reconnect with these people and if they are interested, offer to introduce them to startups in the VC's portfolio as possible hires. Hiring is a key value-add that VCs are expected to deliver, and yet it's very challenging to find the time to actual source and vet candidates for our companies. Having someone help us fulfill that promise and look good in front of our portfolio companies is a big deal. You can also generate the mini-decks mentioned in part 1) to compare portfolio company competitors, recommend product features or suggest growth strategies for a given portfolio company. Even if none of your recommendations gets implemented, the VC will likely appreciate that you surfaced some interesting ideas and best case scenario, they'll send along the suggestions to the portfolio company and feel grateful to you that they were able to help that company out in some small way.

4) Build the skills you're lacking today. You mention that you don't have any real finance experience. To be honest, VC rarely requires hardcore financial modeling at the level of investment banking or private equity, especially at the early stages. There is a lot you can learn from textbooks, career guides such as Vault, online corporate finance training classes and even public financial modeling templates. If you can speak comfortably about basic revenue multiples-based valuation and key items that appear in a term sheet, you'll already be ahead vs. many entry level applicants in showing an understanding of VC finance.

Check out some of these interesting resources:
FunderClub's primer on convertible notes:
https://fundersclub.com/learn/convertible-notes/

A good summary of what KPIs matter for early stage SaaS companies: http://christophjanz.blogspot.com/2013/12/a-kpi-dashboard-for-early-stage-saas.html

Brad Feld's excellent discussion of term sheets: http://www.feld.com/archives/2005/08/term-sheet-series-wrap-up.html

Version One's excellent summary of what KPIs matter for a marketplace business:
http://versionone.vc/marketplace-kpi/

There are many more, but you get the idea - there are many, many blog entries, books and spreadsheet templates that will help you learn the finance and key metrics behind typical VC diligence, which is far from rocket science.

5) Develop your own public brand. It will be much easier for VCs to buy that you'll have the ability to build a powerful network if you're already well known in the community. Certainly, it takes time to build up a reputation, but you can start investing in this today by blogging, attending or volunteering for meetup groups, tweeting thoughtful questions or responses at experts in a given industry, publishing some of the market analyses you've generated (if you acted on tip #1) and even answering Quora questions! If a VC sees your name publicly a few times in relation to other VCs, themes or conferences that interest them, he/she will start to think positively of you as someone who runs in the same circles and therefore has credibility.

Clearly, these suggestions require a ton of time and effort to implement. But that's the point - if you're coming in from a non-traditional background and don't fit the specific mold of what VCs are used to hiring for, you'll need to make an effort to stand out and signal very clearly, in the absence of resume indicators, that you are qualified for the job. The good thing is, most applicants don't make that extra effort, and so it's actually relatively easy to stand out, even among traditional candidates with stronger backgrounds.