As a DM, this, to me, was mistake 1. Just, because it would be more fun, doesn't negate your responsibility as a DM to the game as a whole and the other players. My reply would have been for the player to stick with the original character that night. I would look it over and ensure it will fit the campaign and at the power level.
I believe it is my responsibility to write an interesting adventure and adjudicate the rules. I don't think it's my responsibility to tell people what they can and can't play within those rules. I believe it's the rules responsibility to make sure the game is fun and balanced no matter what people play.
Besides, my philosophy is "If we aren't having fun, then why are we playing?" It's pretty much my primary rule when DMing. If someone comes up to me and says "I'm not having fun", regardless of the reason, I'm going to work to correct it. It might not be possible. But allowing someone to change characters is easy to do and the least I can do to make sure someone has a good time.
Also, I've tried to say no to it before. It doesn't turn out well. People assumed I'd say yes and didn't even bring their old character or actually threw out the character sheet once they finished their new character. So, the choice was to tell them to go home or allow the new character a couple of times. If that doesn't happen I've had people literally threaten to leave the game if I didn't let them switch characters. To them, character creation was nearly as fun if not MORE fun than the actual game. So, if they played too long without being able to go through the character creation process, they got bored.
Mistake 2 was allowing sources you were not familiar with
Mistake 3 was realizing there was material you were unfamiliar with still allowing the character in play rather than having the player hold off
Oh, I was familiar with every source. I bought every WOTC book on release day. I likely hadn't finished reading through all of them. Heck, I STILL haven't finished reading through most of them. I didn't have time. I wasn't going to have my lack of time cause less fun for other people, however. I normally read through the feats in new books almost immediately, mind you. But I'd forget the details of them quickly if I wasn't actively playing a character who took them. So, when I read a feat, my thought process was normally something like "Oh, that's the new feat from the Complete Warrior. I can't remember exactly what it does, but something about trading attack bonuses for damage."
But even if I made the player hold off(see above as to why that was troublesome in and of itself), I STILL likely wouldn't have looked up the feats even given a week. I would spend the time doing things I'd rather be doing like watching tv or playing video games. I'd show up at the next session and say "Oh, sorry, I still haven't had time to look through your character, maybe next week".
Also, even if I did spend the time to investigate the character in detail(which I did a couple of times when I was bored) by looking up every feat, every spell, every class feature, I'd STILL miss 50%(or more) of the overpowered things people brought into my game. This is because most broken things weren't readily apparent. They'd often involve combinations of feats and class features that seemed benign on their own but only when combined together did their power show itself. I didn't make a habit of sitting down and running a combat for the character to see how they actually worked, so they'd go unnoticed until they showed up at the table. It felt like a waste of time to spend an hour or two looking up feats and trying to get a feel for a character only to be surprised by the character and have to ban it anyways.
Mistake 4 allowing the character and not telling the player that the allowance would be provisional and you reserved the right to a) deny the character or b) require changes/nerfs if the character proved unbalanced.
That goes without saying. There was an unwritten rule that EVERYTHING in my game could be changed by me at any point. I didn't need to reiterate it. Though, my goal was to avoid denying any character or nerfing any rule unless ABSOLUTELY necessary. Simply doing 10 average points of damage more per round than everyone else wasn't something to nerf or ban over.
Sorry, this was on you as a DM and to a lesser extent your player for springing it on you at the last minute. However as the DM, it, ultimately, rested on you to delay the character entering and you ignored the opportunity to have the player to do so which led to the same problem with other players and the game getting out of hand.
I believe my responsibility as a DM really begins and ends at the table. I'm here to run whatever adventure I most recently bought at the gaming store for the 4 or 5 hours we've allotted to play. I don't write my own adventures because that takes too long. I certainly don't want to spend any time going over people's characters to make sure they aren't broken.
I want people to be able to have just as much fun playing my D&D game as they would playing WoW or City of Heroes. One of my players used to play City of Heroes and would make an average of 5 new characters a DAY. The fact that they only introduced new characters once every 4 or 5 sessions was considered a win for me.
Frankly, double checking game math and balance sounds a lot like work to me. As soon as D&D becomes work, it isn't worth it.