I’m going to propose a radically different answer than the ones I see here, and also challenge one of the premises of the question.

Elrond’s plan was to send two powerful elf lords from his house. And Gandalf’s reason for choosing not to had nothing to do with keeping the mission a secret.

The Background

There are three elements to the decision:

• The choice that there will be nine walkers to oppose the Nine Riders of the enemy is crucial, and its significance is unexplained.
• Seven are already chosen: Frodo, Sam, Gandalf, Aragorn, Boromir, Legolas, and Gimli. The reasons are: Frodo because he is the ringbearer; Sam won’t be parted with him; Aragorn because he is king and this is his quest, and his other destination, Gondor, is on the way; Boromir as a representative of men and because he is going to Gondor anyway (darker designs for bringing the ring to Gondor aside); Legolas to represent the elves and Gimli to represent the dwarves, so all free peoples of Middle Earth are represented, as this is the war of the free peoples of Middle Earth against their oppressor, Sauron.
• Merry and Pippin decide to go. They decide this in their own council before Elrond and Gandalf select the nine. And they are determined. Pippin says “you will have to lock me in prison, or sent me home tied in a sack,” and Elrond sighs and gives way in the face of such determination and in the face of Gandalf’s support for Merry and Pippin.

The Decision

Gandalf’s stated argument in support of Merry and Pippin going is that it is better to trust in friendship than in great wisdom or power. In saying this, Gandalf mentions that Glorfindel might be a member of the nine, and does not say that having such a powerful elf lord in the party would be a problem because of his power.

Gandalf certainly has unstated reasons himself, reasons partly unknown to himself. He has heard the Music that plays out the theme of the ending of the age, and this guides his choices, though he remembers it vaguely. That theme led him to insist that Bilbo join Thorin and Company. I suspect Gandalf is similarly guided here. Though he cannot foresee the rousing of the Ents, the death of the Lord of the Nazgul, or the rescue of Faramir, he sees that Merry and Pippin, especially in their qualities of loyalty and friendship are crucial to the threads that will weave together into the eucatastrophe that will bring down Sauron and those who are beguiled by him.

Elrond sees true, but less far. His misgivings are appropriate to Pippin’s dropping the stone that alerts the Balrog and looking into the Palantir, alerting Sauron. These two errors are resolved, first by the sacrifice of Gandalf the Grey and second by the awakening rightful power of Aragorn the King.

The Secrecy of the Nine Walkers

The question states that Saruman knew Gandalf’s power, and, of course Saruman would tell Sauron. This statement seems to me - no criticism - to be coming from a bit of a D & D power vs. power viewpoint. Close reading of the text reveals three points that show things are not that simple:

• Saruman knows that Gandalf knows where the Ring is, and if he learns that Gandalf is traveling with Hobbits, he will guess the Ring is with them. But he does not know that Gandalf carries an Elven Ring, nor can he truly understand Gandalf’s purpose. His own lust for power blinds him.
• Saruman is not fully in league with Sauron. If Sauron gets the Ring, he will be a willing vassal. But if Saruman can get the Ring himself, he is a rival likely to be able to defeat Sauron. If the Ring is discovered on this side of the Misty Mountains, that is the greatest risk. Even on the west side of the Anduin, where Merry and Pippin are taken, the conflict between Saruman and Sauron over the ring, combined with their unwillingness to trust their own lieutenants, keeps Merry and Pippin alive and Sauron and Saruman in conflict. If Saruman locates the Ring on this side of the Anduin, he is not going to tell Sauron.
• Gandalf knows that, though they start in secrecy, secrecy will not last forever. Gandalf knows his presence is revealed, presumably to Saruman, early, when the wolves attack before they ascend the pass on Celebdil.

Thus the secrecy of the Nine Walkers is a temporary thing, and not a reason for an Elf Lord of great power, such as Glorfindel, not to be in the party.

The Mysteries That Remain

Two points remain mysterious. Why was the total number of nine so crucial? Once Merry and Pippin decide to go, why not boost the number to ten or eleven, and throw in a couple of trusty Elf Lords, always good to have around in a pinch? And why keep Legolas? If the Elf Lords of Rivendell are so much more powerful (and Glorfindel, who killed a Balrog, certainly is), why not swap out Legolas as the representative elf?

A Theme Revealed

There is no solid answer to this question within the canon. But I am going to propose an answer that makes sense based on a recurring theme throughout The Lord of The Rings. The theme is that this is the end of the Third Age, and also, therefore, the beginning of the Fourth. This war is between the Free Peoples of Middle Earth and their potential oppressor in the Fourth Age, Sauron.

The First Age of Middle-earth told the story of the Elves who left the West to pursue Morgoth, and ended with the resolution of the Oath of Feanor, the guilt, repentance, and valor of the Elves and the acts of the Host of the Valar. In that time, the arrival of men played a crucial role, but their numbers were small and the focus was on individual leaders and heroes.

The Second Age of Middle-earth was the age in which both Elves and Men thrived together, and Sauron played out his deceit and then was defeated, first losing his beautiful, beguiling form and then losing the Ring in the Last Alliance of Elves and Men.

In the Third Age, the Valar and Maiar have withdrawn from the world, with few exceptions. Their role is hidden and they do not use their power. The Elves who have seen the Undying Lands are also fading. The Elves of the West are not part of the Fourth Age, and so do not have a significant part in the ending of the Third Age.

The peoples of the Fourth Age will be men and hobbits and wood-elves and dwarves. And thus it is these people and not the representatives of prior ages, who are represented in the Nine Walkers who are the predominant force against Sauron.

This makes sense of the need for just nine - a mystical hint to Elrond and Gandalf that a higher destiny is guiding the creation of the Nine Walkers to fulfill Destiny - and the lack of a powerful Elf Lord from the First or Second Age in the tally of the Nine Walkers.