That Lear (Garrett) is electric start, so no reason there.
I like the idea of the gas backflow thru the cooling turbine, but suspect gases after shutdown, even without cooling flow shouldn't be a problem... but then air into cabin never mixes with bleed air anyway....
The Old/Bold rule and reason to do it like that is cuz it's in the checklist that way. I'm rated in the Lears, but do not remember the philosophy, it being 20 years since flying one.
Call FSI and ask. They will get you an answer no matter what.