Someone correct me if I am misremembering, but doesn't Psi Training happen AFTER you muster out of a career and not before you enter a career?
Bingo.
I believe it should be hard to find. I wouldn't allow it in chargen.
Finding the Psionics Institute is a reward--a character reward. It should be the prize at the end of a long quest.
The quest rules are in the RAW, and they're meant to be used after chargen.
I would let a player know--the one really interested in checking PSI--that the older the character gets, the lower he's apt to test for PSI. That way, he can make a decision about the number of terms he wants his character to achieve.
It's kind of a balancing effect. If a character goes one term, he's young when he gets out and starts adventuring, but he doesn't have many skills or any skill of a high level. But, he's the most likely to test high for PSI, if an institute is ever found.
When he finds an institute, which he will do at some point, he'll have few and low skills, but he'll balance that out with the higher PSI rating.
OTOH, those characters who go through many terms will have more and higher skills but lower PSI ratings due to their age.
I'd put a character through chargen, then I'd turn his quest for the Institute into the adventure. That's your PULL. Finding the institute. You can work in other adventures as he moves along his quest.
As for searching for an Institute, remember that it's impossible to check week after week. You can't roll 52 times, once per week, in one year. The reason is because there's travel time between worlds.
With all other factors resolved, the best a character can hope for is 26 checks per year, one per planet per week (with a week to jump to the next world).
But, that's a pipe dream. If the character owns his own ship, he's got to fuel it and pay for other expenses. He's not always going to have a full hold, and his contracts are not likely to take him t the next most likely world to have an Institute.
If the character doesn't own a ship, then he's paying A LOT of cash, jumping from world to world, looking for the Institute. Even traveling Mid-Passage, that's 8,000 credits a pop, when not every trip will get you to a candidate world where the rules allow you to check for an institute.
I think that finding a PSI Institute is too important to skimp on the rules and allow a character to find one during chargen.
When does a character have time? He needs leave if he's in a military force, and that leave has to be on a qualified world. He's got to have an entire week to himself. Even if the character owns his own ship, he's got crew to pay, a ship payment, fuel and consumables, berthing fees. And, when he's on a world, he's not making money. Usually, a ship spends a week in jump and a week on the world, but the crew is working when they're on the world. They've got to unload cargo, load new cargo, meet and work with passengers, locate new business and work with the broker. Sure, the guy may have off a few hours here and there--maybe even a couple of days--but he doesn't have an entire week to search for an Institute.
I strongly urge you to not allow this in chargen and make it part of your game.
I bet it will be a lot more fun for you and your group to do it that way, anyway.