There are differences in feel to the player between cork and felt. Cork tends to admit a bit more key bounce than felt. That bothers some players.
It is typical to utilize the same felt thickness for key tails and bend the key tail to remove lost motion. There is of course a right and a wrong way to do that properly. If you do it right, it's pretty easy to remove a small amount of lost motion and keep the "look" of uniform key tails intact.
I used to use cork, but have moved to always using felt now. For me it is a better feel and a better look.
Felts for Pearl flute
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