I've been watching the mooring buoys this season because of the thread here about various tying methods. I'm sure this will change during the upcoming holiday wkd, but so far I haven't seen all of the buoys occupied even on Saturdays.
Yes, The Boardwalk has a dock & dine program, but they have very few actual docks, so the chances of getting in there on a Saturday are slim.
Put-In Bay Marina is the former Ladd's marina - closest to the monument. Sadly, due to financial issues, Axel & Harry's and the High Tide didn't open this year.

However, the marina operations are separate and they're running full-steam. So much so in fact, that they have appeared to be nearly full each wkd.
The old Duggan's Marina (between the Jet & Public C Dock) is Fox's Marina. They raft, but I've never stayed there, so I don't know if it's mandatory or not.
Public Docks A & C share management, B Dock runs separately. The facilities, policies and I think even the fees are all the same. The public docks don't always deserve their reputation. Yes, they're full of loud drunks, but the dockmasters plan the dockage very carefully. If they know that you'll only be there for a few hours, they'll make an effort to make sure you don't get rafted to. Of course, there's always the risk that they will underestimate the demand, but I wouldn't write them off completely. As always, a tip and clear communication about your expectations help considerably.
I'm with you by the way, on thoroughly enjoying Put-In Bay on weekends. It's my favorite destination in the Great Lakes. Having said that, have you ever considered joining The Crew's Nest? Their docks are ridiculously well-managed by a team of hard-working, polite and competent young adults. They have no mandatory rafting and more docks than any other marina on the island. Yes, there's a financial hit for dues, initiation and a wait-list for boating privilages, but they also have great facilities and service. (also watch for specials - they have offered discounts on the initiation fees at the Cleveland boat show for example)
The Put-In Bay Yacht club is ILYA affiliated, which can come in handy elsewhere, and they also accept non-members when they have room to do so. Miller Marina is under new management as of last season, and they accept advance reservations, so that's another number to keep handy.