Based on the exchange you've had with the previous repository maintainer, the previous maintainer never transfer ownership of his copyright. This situation is virtually identical (from a copyright and licensing perspective) to forking someone else's freely-licensed repository without coordinating directly with them. The original owner holds copyright on the original code, and you have permission to make your own modifications, which will be under your own copyright.
The only difference here appears to be that the Github repository has been mechanically transferred from being listed under the previous owner's account to being listed under your account. This by itself appears to have no copyright transfer implications. It is merely an easy way for the previous maintainer to signal, "I don't care about updating this project anymore, but this guy does." Having the repo listed under your account doesn't mean anything in copyright terms -- the license text always allowed you to copy the code and post a new copy under your own Github account, which is nearly identical to what has happened here.
Unless the previous maintainer has explicitly authorized a transfer of title making you the new copyright owner, you do not have the right to remove the copyright notice. If you would like to do so (and if you would like the unambiguous ability to completely re-license the entire work in the future) you should ask for the maintainer to completely relinquish copyright to you, which he may or may not wish to do.
Assuming you have participated in a transfer of title and you are now the new copyright owner of the entire work, then, yes, you can remove the copyright notice. this is trivial to see, because the previous copyright holder no longer has any legal standing to take legal action against you, since only the current copyright holder (who is now you) can take legal action against misuse of a work. This hardly seems like a polite thing to do (versus simply adding your own copyright notice as an addition to his), it appears to be a legally valid thing to do, assuming you own the copyright on everything.
Note that outside of the U.S., some jurisdictions have "moral rights" that can never be surrendered, and those moral right may include the right to be acknowledged as an author of a work. It's possible that removing an old copyright notice may run afoul of this moral right in certain jurisdictions, but I'm not sure.