I believe that's butyl tape, not caulk. Over time it expands/contracts and the edges get brittle but what's underneath what looks like a corner trim piece in your pics should probably still be good. I would take Grumpys suggestion and trim off the excess and seal it if anything just for looks.
x2 on OSI Quad. Just be careful when you use it. It is thicker than "normal" latex and you have to have a strong pull to keep it even. I would try a 3 to 4" practice run off the TT. I have found that using hot water with a rubber finger helps to smooth out any lumps or pockets.
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