That there are statements which might be true but we don't know how to prove was always the reality in mathematics. That this is not a temporary reality but something we're doomed to was thus not a huge revelation. Definitely not something I would expect anyone to lose sleep over.

Also, is it true that constructive mathamtics actually escapes Godel's theorems? My understanding was that any system strong enough to contain regular arithmetic is subject to it. Either way, constructive math is more limited in what it accepts as proofs, so there will be more, not fewer statements that are true but unprovable with such a system.