From Chapter Fourteen of Philosopher's Stone:

“Oh, come on, Hagrid, you might not want to tell us, but you do know, you know everything that goes on round here,” said Hermione in a warm, flattering voice. Hagrid’s beard twitched and they could tell he was smiling. “We only wondered who had done the guarding, really.” Hermione went on. “We wondered who Dumbledore had trusted enough to help him, apart from you.”

While Hermione might be totally making this up, it is possible that she has correctly identified the point, namely, that Dumbledore trusts certain people. According to this we can surmise that those who helped protect the stone were the teachers that Dumbledore particularly trusted.

Because had any more professors been involved, there would have been more than seven obstacles. This is Harry Potter. Everything is about the number seven.

I'm not sure if this is what you refer to as the obvious out of universe answer. But even in-universe, characters know the power of seven, for example the deliberate number of Horcruxes, or the ways of scoring in Quiddich.

Another answer is that Dumbledore wanted obstacles that would stop Voldemort (if he had returned) but would not stop Harry (and his friends). Or, as the inspiration for the question says, "A first year should technically be able to get through with the help of some friends." By restricting the professors to those that knew Harry personally, he could make sure the challenges were things that Harry, specifically, could overcome.

Possibly limited to heads of houses and other important persons rather than all staff.

Fluffy - Hagrid (trusted person)

Plants - Professor Sprout (Herbology), Head of Hufflepuff

Flying Keys - Professor Flitwick (Charms), Head of Ravenclaw

Chess - Professor McGonagall (Transfiguration), Head of Gryffindor & Deputy Headmistress

Potions - Professor Snape (Potions), Head of Slytherin

Mirror - Professor Dumbledore, Headmaster

It's possible that the traps were meant to serve multiple uses...

1. Allow Dumbledore through, without too much hassle.

2. Slow-down - rather than stop - someone trying to get the stone, so that Dumbledore could be notified (wards) and had time to intervene.

3. Serve as test for Harry Potter (whom Dumbledore knew eventually would have to be the one to stop Voldemort) and his friends. If so, it makes very much sense to only use the teachers teaching 1st year subjects. (In addition, Hagrid conveniently gave Harry a flute for Christmas to put Fluffy to sleep.)

4. Make sure no one teacher could easily disable all traps using their own expertise.

5. Test the teachers' loyalty, by seeing how difficult their traps were (eg. Quirrel's troll).

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If Dumbledore just wanted to keep the stone safe, why use a simple locking-charm that several students (including 1st year Hermione) could bypass?

If Dumbledore wanted to keep the stone safe, why not bury it under the stone-floor of his personal quarters (accessed through his office, protected by a gargoyle) and use many layers of wards?

Why tell all the students at the welcome-feast, in a way that almost guaranteed some students would try to find-out what was hidden?

The only real safety-measure was the mirror, as someone who wanted to actually use the stone, would spend hours trying to get to it... assuming they didn't also succumb to the magic of the mirror, and "wasted away in front of it".

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So I would think everything but the mirror were just slow-down an intruder, and to test Harry and his friends. If Dumbledore wasn't away, I'm sure he'd observe Harry's journey through the traps - maybe he did.

While I think that the other answers provide quite reasonable answers, I think it's also important to recognize the rather special relationship Harry and Dumbledore have throughout the series, as well as the connection between Harry and Voldemort. Dumbledore makes a habit of spinning lots of plates at once, making things seem disconnected, until (unsurprisingly given the name of the series) it turns out to all be about Harry - Snape's revelation in book seven is the epitome of this.

All that is to say that:

Voldemort will make it through any security system, so Harry needs to be able to as well.

Harry, at this point, is the one thing that can stop Voldemort in the end. Dumbledore knows this, and intends for Harry to get involved in order to stop it. Dumbledore can't do it himself because:

1. He doesn't want to show his hand. He's still amassing power against Voldemort, and doesn't want him to come out of hiding too early.

2. He needs Harry to start believing in himself and his ability to fight against Voldemort, if he wants Harry to be an effective fighter.

Furthermore, Dumbledore doesn't want it to be someone else besides Harry, because he knows that Harry is still protected by his mother, and will never just give the stone away.

It's pretty clear that Harry was meant to get through these challenges - they're basically tailor made for him and his friends. The only reasonable explanaition for such a personalized set of tasks is that they were meant to be that way.