How to effectively patch/recaulk small crevices between bathtub and bathroom tile?
That is not a bathroom crevice that is a canyon.

You can not stop water from flowing in a canyon unless you build a big ass dam.
But before that WTF is going on in that tub?  HOLY SH!T was my reaction.   Nice tile, good quality tub (I can tell its cast iron) and egregious small mistakes.   WOW!!
Let me give you the John Madden teleprompter rundown.
#1 - ILLEGAL USE OF CAULK, 15 yards, HOME DEPOT CARD TAKEN AWAY

#2 - What?   What?   What?   You have a gigantic crack, a canyon and that canyon is on rails that contains a large river and it is just wide open?   This is like Jerry Rice vs SD in the Super Bowl.   Touchdown water.

#3 - I see what you did here.  You went Deacon Jones on the water.   I will put so much caulk above this crack that it will get a headslap on the way down.   Caulking above the crack only works in Australia.   That is a DIY fact.

#4 - The Mike Martz.   Your tile and tub is Kurt Warner.   Please please coach give me some protection.   Wait they are blitzing 6.   Well we will run 5 deep patterns and see what happens.    Well you don't have time for that you have mold.

Summary:  What must happen, must happen.  This is not me speaking to you, it is your tub, your walls, your sanity.

Tub door shall be removed.   I would bury it.  It wasn't installed right.   Get a rod and curtain that will keep the water out of 90% of the places you are having issues with.  This door is funneling water EVERYWHERE.   It is the Colorado river on steroids.

Grout all of the large cracks between tile and tub once door is removed.

Caulk all of the small gaps (last picture) between tile and tub.

Realize it will take 1 hour to take that door out and 45 mins to grout and 20 minutes to caulk and you probably have a tub running at 95%.   It isn't that bad you just have that door sabotaging right now.