so I hired someone that I don't intend to hire again to replace it. .... In the course of correcting his mistakes...
Well it is true we're really scraping the bottom of the barrel for electricians these days. However, honestly, this sounds like something else entirely: it sounds like a person who knows a little bit about electricity and has... well, we used to call it "sophomoric", but a pair of scientists named Dunning and Kruger made a more in-depth study.
I'm afraid I'm not referring to the electrician :)
It is readily apparent that a) you're seeing a lot of stuff that is "New to You"... and b) you're doing a lot of random stuff that seems right, but you don't actually know. At least, you are asking...
Receptacles must match the breaker.
This is NEC 210.21.
15A receptacles belong on 15A circuits.
20A receptacles belong on 20A circuits.
There is an exception allowing 15A receptacles on 20A circuits if there are 2 or more sockets because 15A devices are internally rated for 20A pass-through. Not 30A.
30A receptacles belong on 30A circuits. That is it, no crossovers, no exceptions.
That receptacle you're familiar with that you've seen everywhere, has 2 sockets for purpose of the above.
You will need to change each receptacle and spur on that circuit to a 30A receptacle. You may use a NEMA TT30, NEMA 6-30 or NEMA 14-30.
"I don't want to do that, though!"
Wire is always allowed to be larger
Anytime Code requires wire be a particular size, you are always allowed to use a larger size. There is no penalty for this, except the practical difficulty of fitting the larger wire to terminals.
The exception is that if conductors are enlarged, grounds must be enlarged in proportion, but that works fine with #10.
What this means is you're in perfectly good standing to have #10 wire in a 20A or 15A circuit.
Enlarging wires is often done to compensate for voltage drop on a long cable run, or because old wiring (e.g. from a decommissioned electric dryer or range) is being re-purposed for new circuits.
The circuit size is the smallest wire size in it.
In residential, where the Tap Rules are generally unavailable, the circuit size is defined by the smallest (in-wall) wires in it. If a circuit has a mix of #10, #12 and #14 wire, it must be a 15A circuit.
If a circuit has a mix of #12 and #10 wire, it must be a 20A circuit (15A is acceptable).
It can only be a 30A circuit if every single wire in the walls is 30A (#10 copper).
Generally anything with #10, #14 and #12 is defined by NEC 240.4. It cuts out some really weird exceptions for certain motor and welder loads, but this is generally not something you'll encounter on receptacle and light wiring.
Multi-Wire Branch Circuits are a thing
AKA shared-neutral or "Edison" circuits (for whom we have to 'thank'). These are +110V and -110V (at least in Edison's day) with neutral in the middle. Only 3 wires are needed + ground, but it has the performance of 2 circuits. The neutral handles only differential current.
These must be phased correctly to avoid overloading neutral - the 2 phases must be 240V apart, not 0V apart. They cannot land on a tandem. To protect maintainers, they must have "handle ties" or be a 2-pole breaker.
The difference is that handle-ties do not provide common trip and most 2-pole breakers do. However MWBCs do not require common trip if they serve only 120V loads. (yeah, they can serve 240V loads also, and then, they need common trip).
MWBCs also require that neutral be pigtailed and do not use the device itself for splicing neutral. Neutral must remain continuous even if the device is removed.
Cleaning it up
The only mistake you have described is lack of handle ties on a MWBC. That's a common error, I'm not going to judge the electrician incompetent over that. Hopefully they are on opposite poles (240V between them).
Everything I've read tells me that, as far as the math goes, that's okay because the neutral is still only handling 30 amps, which 10 AWG can handle.
You didn't read that in NEC/CEC. You might get away with it on the #10, but what about the #12 or #14 beyond the #10? You haven't fully mapped the circuit and don't know what all the wire sizes are.
If you have mapped the whole circuit and found nothing smaller than #12, then you can use 20A as long as there are 2 sockets on each leg (or the 1 socket is 20A). If you have 15A on one leg but the other is all 20A wire, then you can use a 15A and 20A breaker with a handle tie. I have one of those!
I'm worried that the breakpoint is too high and and that I should be using two 15 amp breakers with a common trip.
Correct, that is the problem, but common trip is not required for MWBC.