There's rent-a-buddy mode, where DM is truly acting as a quite experienced site familiarization person for one other diver. In that case there should be full buddy checks, hand signal review, sticking close to the buddy.
There's charter group guide. In that case DM is helping with equipment setup on the boat (unless specific divers are don't-touch-my-equipment -- their choice, ask in advance).
DM then does the site briefing -- general path being followed, any standard landmarks, special site safety recommendations (e.g., on a deep dive, spending an extra minute at half depth on the way up, if it's a wreck discussing whether there are any "safe" penetration areas), recommended turn pressure guidance.
DM checks to see if anyone is buddyless, and if someone is alone understand whether they desire a buddy or wish to act in a solo mode (in solo mode a good DM still will try to keep an eye on the diver, if the diver is close to the overall group, do a periodic gas check signal perhaps, but it's a very loose "leash").
If a diver is buddyless, doesn't want to be, DM pairs the diver up (ideally there's another single diver, else a triple [not optimal, but better than an unintended solo]).
DM asks if everyone has done their buddy checks (don't have to actually go to each pair, just a polite question to the group, a reminder).
A discrete valve check/question (gas fully on?) when divers are about to enter the water.
Camera handoff from the boat to the divers.
Underwater, DM leads, points out interesting formations, flora, fauna.
It's much easier when there are a couple of DMs with a group, then one DM is lead, another acts as sweep at the back.
One thing I'd personally be leery about is if a person indicates they're alone/solo, the dive is deeper than 25-30 feet, and I don't see the diver carrying redundant gas (a pony). In that case the diver should be paired/tripled, doesn't matter if I think they said they want to be solo -- basic safety at depth requires access to backup gas. (People are ultimately personally responsible, but as DM you want to catch obvious whoopses, and solo deep without backup classifies as an obvious this-isn't-right)
Concerning the other thread, lack of communication does happen, but if the "solo" diver was indeed going deep, didn't have backup gas, that one item would be a flag against the DM -- other attributes would be the diver's responsibility (in other words, DM should catch the no-backup-gas issue, but the client also, as a responsible diver, should not be attempting the dive, or should be explicitly running a shallow profile, and state that up front).