> I don't really trust it
REAL ID is a bare minimum. It shows that you at least have legal residency.
FWIW it's trusted by DHS so that's all that matters for your usecase or assumption.
If you are worried about the risk of being hauled by ICE, then you should get a REAL ID.
> it requires me to send my existing passport for several weeks
Last I remember, you can do it in person.
Here are the passport offices - https://travel.state.gov/content/travel/en/passports/get-fas...
> In the event of a government shutdown
The US Passport Office remains open during the commonly termed "government shutdowns"