I’m pretty sure that used to be the case, and may still be.
A friend used to live just south of Patrick AFB (like within sight of the south base boundary), and you could often see the T-38s there, when driving by on A1A. They were like the astronauts’ sports cars.
I see them flying all the time around Ellington field. A most puzzling waste of money. Astronauts, by definition, travel beyond the atmosphere where flaps and ailerons are useless, and yet they're required to maintain something like 100 hours per month in the T-38 cockpit, which doesn't remotely resemble the shuttle flight deck or the Apollo capsule interior.
What part of the T-38 experience can an astronaut apply to spaceflight? The usual rationale is "it's a shuttle trainer", but it handles nothing at all like the shuttle, and besides, it's use predates the shuttle. NASA has been using it at least since the Gemini era - for what? The Gemini astronauts and most Apollo crew were already skilled fighter pilots.
I think it’s mainly overall flying skills maintenance.
Training often simulates mission activities or mindsets in correlative capacities.
Repetition itself is very powerful, especially of pilot awareness skills.
What part of the T-38 experience can an astronaut apply to spaceflight? The usual rationale is “it’s a shuttle trainer”, but it handles nothing at all like the shuttle, and besides, it’s use predates the shuttle.
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I believe some of them were modified to handle just like the Shuttle.
During the shuttle era, I believe the T-38’s had their controls modified to simulate shuttle controls and feel.
It’s one of the perks of the job.