The first line of a feat is a description, not proscription. It describes roughly what the feat is about; it isn’t the place for specific rules and limitations on it.
Those are part of the Benefits section. Thus, when we look to the rules for Psicrystal Affinity, we are unconcerned with that first line, and its use of the past tense. We are interested in the Benefits section.
That says the feat “allows you to gain a psicrystal.” This is present-tense, and worded specifically as an ability you now have, to gain a psicrystal.
Unfortunately, almost nothing about the process is described. We aren’t given an action or timing for how long it takes, we aren’t told what, if anything, it costs, we aren’t even given any limitations on its use—despite the fact that the game repeatedly refers to “your psicrystal” as if there can only be one.
Ultimately, with rules this sparse, it is necessary for the DM to make some rulings on some of these things. Your first psicrystal, it seems pretty clear, is made as part of the process of leveling up—the rules are intentionally vague on this to give your DM room to decide what that looks like, but it’s generally assumed it’s not literally “ding!” and a psicrystal pops into existence. Rather, the Player’s Handbook and Dungeon Master’s Guide talk about how maybe the DM should expect some downtime, some training, etc. If nothing else, it’s assumed that the character has been working on the abilities of their new level during the previous level, in the background. The level-up is just the “Eureka!” moment where it all comes together and suddenly works. So the first psicrystal happens during that.
What about psicrystals after that? You have a feat that “allows you to gain a psicrystal,” and nothing says you can’t gain a psicrystal after you already have one. There’s a lot of circumstantial evidence that a given character isn’t supposed to have more than 1 psicrystal—again, see everything talking about “your psicrystal”−but this, too, has to be a matter of ruling. I would expect most DMs would—and should—say you are limited to having just one.
But if that one is lost or destroyed? You have the ability to gain a psicrystal. You should (be able to) use it. As a standard action? No, I wouldn’t recommend that—yes, there’s a RAW case for it, but this seems a clear omission. But as a process requiring a moderate amount of downtime? That seems reasonable. I’d probably call it one block of 8 hours’ work, myself.
What doesn’t seem reasonable is limiting the psicrystal to 1, ever. Contrast Psicrystal Affinity with, say, the Exalted feats from Book of Exalted Deeds—that book goes to considerable lengths to say if you lose the feats, you just lose them and can’t get them back. That’s necessary because dead feats aren’t a normal part of the system. The rules for psicrystals don’t make any such explicit statements, and they are necessary here. Reading them into the feat, on the basis of the tense used in the description (or on any other basis), is a considerable twisting of the apparent intent for psicrystals, and a DM should—at the very least—make this painfully clear to the player before Psicrystal Affinity is selected.
On the other hand, gaining multiple simultaneous psicrystals, particularly as a standard action, while on much stronger footing, RAW, is also a considerable twisting of the apparent intent for psicrystals. A player likewise should make painfully clear that this is their expectation and plan and not spring this on a DM.