[ Ref. 171011 MMD, Standard Pneumatic Nightmare ]

How horrid!  Sad to hear it.  Is it acrylic caulk (decorators' caulk)?
That would almost fit the bill for "looks the same" (goop laid down
with a caulking gun) and "more breakable, less rubbery".  It's like
white glue with a lot of chalky filler, and gets brittler over time.

It seems to me that ordinary paint stripper (the less-evil modern DIY
kinds -- ironically, the ones that are feeble on tough old paint!) is
able to re-melt this so it can then be scraped out.

It's very hard to get a wood surface so clean that you can stain it
(which is what I was trying to do -- was working on abused Victorian
woodwork, not a piano) but it does seem that shellac-based finishes
will bond to the wood after this treatment if the stripper is cleaned
off really thoroughly.  I used Paint Panther but I suspect other
brands are now much the same.  They don't seem to loosen silicone.

Speculative, alternative thought: If you heat a smooth, flexible
scraping tool, e.g., a palette knife, can you then slide it under
the valve?

Lester Hawksby (sadly bogged down mending a lot of things that aren't
instruments).