Just a thought about this Bt icon thing: have any of you installed specific launchers, and if so, which? All info including negative, welcomed.
I don’t experience the problem and never have [edit: so far …]. FP3+ / Stock Android 3.A.0129. Stock launcher. You’d have a job finding more standard than me. Anything for a quiet life! 
To begin with I hardly ever turned Bt on except for the occasional file transfer, but since December 2020 I usually leave Bt on all the time as it’s required by the Covid app. So I cover both those use cases! I’ve never found Bt off, when it “should” have been on, or vice versa.
Airplane mode ON turns Bt off (naturally). When I turn Airplane mode OFF, Bt comes back on as expected.