I completely agree with manassehkatz about getting a Nice Big Panel. We have a 24-space in our shop fed by #6 THHN wire (65A, smaller than your 85A) which more than half full and we haven't even really started installing 240V tools. You can support a lot of stuff with a feeder as large as yours. And before you say "EVSE is gonna soak it all up" lol not nearly. Sensibly sized EVSEs don't draw all that much, and "bonkers" sized EVSEs don't run for very long.
The old panel is mounted at eye level, so simply moving the new one higher is not a good option. My plan was to use an 8"x8"x4" deep electrical box mounted high on the wall, drill a hole in the back of it with a protective bushing, and lead the branch circuit wiring into it to splice on new leads which I would run into the new panelboard through a short piece of EMT.
Sure, I do that all the time, it's practically a signature move for me. Not least because I realize I can use 4-11/16" square boxes and those will nicely fit two GFCI or AFCI receptacles or deadfronts. Cheaper than breakers, and easier to tell why it tripped!
But if you're thinking "I'll buy a panel that same size", lol no. Newer panels are much taller, because UL requires much more wire bending space, and "convertible" panels are all the rage (main-lug panels with space left to fit a main breaker). So if you listen to an ounce of our advice about "bigger panel", you're going to have a panel with 6-12" of empty space above the first breaker. You may not need a separate box!
It's perfectly fine to make those wire-nut splices for length inside the panel. Even in Canada it's legal, since the wires are going to this panel.
Anyway you should get at least a 125A-bussed subpanel, since I am deducing (from your "can't fit ground" claim) that you have 1" conduit and 2 AWG wire in it, and it's clearly copper. That is 115A wire, really, so you can pull 115A through it. (Unless it's actually type TW, which would surprise me, then 100A). But for that you need a 125A or larger bussed subpanel. Nothing wrong with 225A bus - heck someday you could flip it and make this the main and the other one the sub.
Did I mention you can power a lot of stuff on this panel?
One supplemental question: Would it be good practice to augment my grounding system by driving a ground rod near this box and connecting it to the grounding bus?
I can tell by something else you said, that you have apprehensions about the viability of "EMT metal conduit as a grounding path". Don't worry. Build it properly and it's an absolutely stellar grounding path - and one of my buildings with 60 year old rusted EMT has "taken the Pepsi challenge" on that, in an incident I'd rather not describe lol, but where the ancient EMT performed tip-top.
But if you want to just run a bare copper #6 outside the conduit from panel to panel, feel free. Just attach it to a lug at both ends and bolt it to the panels with a bolt and nut at each end. You can also come into the panel through a cable clamp in a knockout if you really want to or if you want to cover the whole thing up at some point.
Why #6? #8 is insufficient for 115A wire because of NEC 250.122(B).
The ground ROD won't help the grounding system clear a bolted fault (return a whole bunch of current so the supply breaker rapidly trips). The conductivity of dirt isn't nearly good enough for that.
Ground rods have 2 roles: route natural electricity (ESD, lightning) back to earth, and keep the dirt around your house near the voltage of the neutral-ground bond at your main panel. That way you don't have a 40 volt differential between the actual dirt and the ground on the outdoor outlet on the side of the house. In that sense, the more ground rods the merrier - ideally on all 4 corners of the building. Of course that would cost money, so builders lobbied for two ground rods 6' apart and done.
Eventually, when funds permit, I plan to rewire the entire garage apartment in accordance with current code
Don't lose your mind doing that. AFCI and GFCI protection covers a lot of old sins. Did I mention my trick with the boxes to avoid _FCI breakers?