The "wouldn't get much of an improvement through training " probably refers to their current limits. Goku broke his limits and touched his deep potentials in order to use Ultra Instinct, so what Vegeta said may not be true anymore.
IIRC saiyans do not get stronger in the middle of battles, they get stronger after recovering from wounds. The Ultra Instinct allows Goku to adapt and evolve constantly during the fight. But I don't see how he can defeat Jiren, unless he can pull off something like Ultra Instinct Blue.
