I might do that next playthrough. This one I'm trying to save Alfred, but next time I'm probably going to be a complete asshole, just send people to Iosefka, etc.If you want to get the Wheel early, you can literally just kill Alfred at his second location to get the badge, which unlocks both the Wheel and the Ardeo. This means you can have the Wheel by the time you kill Vicar Amelia (which is what... a third into the game?). But if you want to give him the summons for it instead, I get why.
As for an Arcane build... any weapon that doesn't have a built in Arcane damage split (blades of mercy, Ludwigs, Tonitrus, etc) gets a 100% damage conversion the moment you stick an Arcane gem into it. That's something to consider when you go into an Arcane run, as it makes some weapons (Saw Cleaver, Saw Spear, Threaded Cane) viable options. That said, the Wheel and Tonitrus are still, by far, the best arcane weapons. The Wheel's buff can become MASSIVE if you have a good chunk of arcane and Tonitrus gets a buff not only in it's unbuffed state but a MASSIVE one in it's buff state. Stay away from Ludwig's, as you need a ton of high quality gems to get close to the DPS of Tonitrus.
Seriously... if you want to kill a boss, Tonitrus is basically always your best option. Most bosses don't resist bolt well and the buff at minimum stats is equivalent to bolt paper. Toss in a bunch of arcane and suddenly that buff is way better and VERY nasty. So my advice for an Arcane build is to clear Old Yharnum, get kidnapped so you can get to the Gaol early, and get Tonitrus early on. Alternatively, kill Alfred right after Vicar Amelia so you can buy the Wheel.
The Tonitrus is pretty nice even when not built up on Arcane. It slaughters the denizens of Cainhurst ... except Logarius himself. I kind of hit a wall with him tonight, so I may do him after the Forbidden Woods. The first time I fought him in playthrough 1, it was already after Rom.