There is a difference between a silent protagonist and one who simply isn't voiced.
Half Life had a silent protagonist. Gordon Freeman never speaks to anyone. People just talk to him (or perhaps it might be better to say "at him") and he follows along. New Vegas does NOT have a silent protagonist. You interact with other people and have conversations with them. It's just that your lines are not voiced. Now that we have that distinction out of the way, let's proceed...
That being said, no, I do not think it would be a good idea. Considering the possible variations in the character you can make, one-voice-fits-all would not work out. You need a more "focused" protagonist for that to work. Mass Effect made it work by giving you a 25-35 year old soldier. Sure, you could customize him to a degree, but you stayed within certain parameters. Dragon Age 2 is also going with a voiced protagonist, but now it's forced to limit you to a Human of 30ish years. Dragon Age 1 let you make anything from a 18 year old dwarf lad to a 90-something wizzened old human mage, but the voices for such characters would be quite different (not to mention the other possible characters), so the protagonist wasn't voiced. It's a tradeoff, freedom and choice VS voiceovers.
One could argue that the developer could just make multiple voice sets and then just let the palyer choose from, say, 4+ voices (per gender), but that's just not an option in this case. Making several sets of voiceovers for the player to choose from would be a logistical nightmare from a development perspective. Also, it would further hamper the developer's ability to edit content and all the voice work would have to be done after everything else is done. Sure, you might argue that having all the other NPCs voiced would also make editing quests a problem, but it's different when you need to drag a single voice actor to the studio to edit some dialogue or when you need to drag 10+ to edit some protagonist dialogue.
Voiced protagonists work in more linear games. The more freedom is given to a player, the less feasible the whole thing becomes. New Vegas is a very open game and fully voicing the player character would be waaaaay too much work.