Is there any way to find every member of an alliance without exhaustively searching through the players list?
If not, why are we prevented from finding out members of any alliance quickly?

You are not prevented from anything. Just limited to the restraints of the game. Why do you want to know this info? If it's because you have some players you want to put on a hit list then you have to search them out. The same as anything. It's part of the game is all.

Oh, alright then.

It can be done but is time consuming. If you do decide to search through all the systems in each galaxy, keep an eye on where you get to. The anti bot protection will cut in at some point and will take you back to where your main world is. So, annoying! lol.

And that needs fixing. The game has to bring us back where we were before the bot interrupted.

And don't forget that harsh white that comes in for an instant between each page. Your eyes beg to be put out of their misery after 50 or so systems pass by.
And the bot protection is annoying as well, I agree, but I see why protecting against scripts is a concern in such a game.

if you are trying to map out galaxies, looking for specific groups, such as actives, or a specific alliance, you can hit the arrow keys on the galaxy screen, and it should page you through more quickly. I have the problem with the flashing white screens, on google chrome, but not on internet explorer. this could just be me though.

p.s. use a spreadsheet to keep track of locations, there are ways to sort the results based on name, and such depending on your software

Now that I think about it, using a spreadsheet is becoming more and more attractive of an option to organize my findings.
I'm just glad that Standard has way less systems than XTreme or Speed. And a lower planet count cap, for that.

It is the upside of that server. Pros and Cons in all of them. You have your notes under the player menu, you can store information relevant to your needs there as a quick reference point.