So one thing I didn't notice, and certainly could have missed, in the original post is the "why?" Which is to say, why does the player want to play a mute? I feel like the players motivation should be a big factor in working all of this out.
If the reason is just that the player thinks it would be an interesting "quirk" I think it'd be fine hand-wave this away with any one of a nearly infinite number of fantasy contrivances. If on the other hand the player has a genuine interest in playing a character with rather serious disability in their chosen field, then I see no reason to do anything more complex than hand them a spell list that contains no verbal component spells.
It would be a tough call for me as a DM if I'm honest and something I could only work out with my own player, rather than fabricate some definitive response to you folks here on a message board. My gut says to work with the player to come up with a way they can cast all the spells in a relatively normal, if quirky, fashion. But the more I think about it the more I wonder "what's the point then?" Why play a mute wizard or other spell caster to begin with if it's just going to mean a bunch of time & effort spent between the player and DM figuring out how to bypass it and turn it into nothing more than some superficial coat of paint? A big part of me would rather see player have to put themselves in that characters shoes and struggle the way the character actually would. I don't know, like I said it's a tough call.
As an aside, I've contemplated playing a character like this, only my concept is a Warlock. Of course being a DM this PC is just a dream like so many others... oh to play as a player. Anyway, the idea was to have a character who had been horribly disfigured and incapable of speech and this leads them to make your classic "deal with the devil." The idea was to go Pact of the Chain and then get the Voice of the Chain Master invocation. Of course this can't happen until level three, but I actually don't mind the idea of having to struggle through the first few levels, sort of proving the character worthy of the gift, until finally he gets his voice back... after a fashion.
Anyway, it's a cool concept but I think this is a case where the DM and the player really having to sit down and hash out the why's and motivations behind this choice before they can reach a solution that'll make everyone happy.