That is an old "rule of six" panel, which while grandfathered, is illegal under its grandfathering becuse it has 7 main breakers. Going to five is a good plan.
It is a classic "CH" panel which is a very good industrial grade panel, except that the 3/4" breaker width make non-ordinary breakers very expensive (a trait it shares with Square D QO). That makes it perfect for what you plan.
On your subpanel which would be near this panel, I would get a panel with a main breaker, with an eye toward (at some point in the future) cutting it over to be the main panel. In a subpanel, the "main breaker" is nothing more than an on/off switch, it is OK for it to be larger than the feeding breaker.
I would also get a rather large panel, at the very least 42 space and even 60 or 84 if practicable: because panel spaces are dirt cheap and often even come with free breakers, whereas running out of space is painfully expensive.
I would aim for an industrial grade panel of good repute (one available in 3-phase variants, not Homeline, BR, or second tier brands) and avoid the expensive 3/4" breakers (not CH or QO).
Over time, as you find it convenient, i'd migrate all your 1-pole and smaller 2-pole circuits over to the new panel.
For your garage panel anything would do, but I'd go for the same type as your indoor panel, so you can use some of those bonus breakers. Again it's false economy to scrimp on spaces, I'd go 20-30 at least.
Also, since garage spaces need to be on GFCI, consider getting a subpanel which has a "main breaker" which is GFCI, that way all the breakers in that panel would be protected (at the cost of potential nuisance trips, a big deal if you keep a freezer in the garage).
Ed Beal raises some very good concerns about overall capacity. One problem with these "rule of six" panels is there is literally no main breaker to stop you from drawing more than 150A. So it pays to be conservative.
It's a difficult situation because you have two big loads that operate sporadically - the EV charger and the range. And the A/C as a wildcard.
One thing I might suggest, is feed the garage subpanel from the new primary subpanel. And then move everything but the range over to the new subpanel. At that point the only things still in the CH panel would be a 60A range breaker and a 100A subpanel breaker. Even at max, those two could not overload the 150A service (by enough to matter). This would force your entire house (from A/C to EV charger) to share 100A, but would remove the possibility of an overload. This would also save you the $85 you'll spend on a second 100A CH breaker.
Don't the two hot legs need to be balanced if it's line to line to compensate for the fact that there's no neutral?
No, if there is no neutral then all the loads must be connected across phase to phase. In US terms this would mean that there are no 120V loads that need to be balanced.
I'm not convinced you have two hot legs from a US style split-phase supply.
The Chinese standards seem to be pick and mix of other countries standards. Note that Red=Live, Blue=Neutral, Green/Yellow=Earth is a valid colour combination in Australia.
The four "two-pole earth leakage breakers" (tengen DZ47LE-32 C16) are marked with N for Neutral at one connection.
I would therefore, naively perhaps, expect the blue wires to be neutral and to have zero or very low voltage with respect to the green/yellow ground wires.
I don't see any normal bus connectors, it seems the installer used red wires to construct buses live and instead of the neutral terminal that would be found in a UK consumer unit.
CE marked C16 circuit breakers should be 16A breakers with a "normal" sensitivity range.
The blackening of the top screws of the old breaker is also concerning. Is too much resistance on the wire or screws/breaker a possible cause?
The clamping screws were not done up tight, the cables were not inserted correctly or the switch was passing too much current. Or all three.
Example, for comparison purposes, of an old and unusually neat UK equivalent showing copper bus bars at bottom connecting live feed to single-pole breakers, terminals at top for neutral and ground (these terminals are sometimes also called bus-bars). Note the copper bus bars normally have a plastic shield, and normally there is a cover that shields all the wiring with holes for access to the breakers only.

- Image by RF Lighting
Since China is not the UK, there may be entirely different exemplar Chinese installations that bear little relation to this. The question shows an installation where every breaker and switch has two poles. This means there is no need for a neutral terminal connector strip. However the neutral wires linking adjacent "earth-leakage breakers" form an improvised bus.
Best Answer
First, you say your service is 100A, but those "main supply" wires look awfully small for 100A. They look a lot like #6 copper, which is only good for 60A tops. I would make sure the wires are appropriate for the main breaker. If they are inadequate, either upsize the wires or downsize the main breaker.
This ... I hesitate to use the word "service panel" ... is still supported. Bryant was bought by Cutler Hammer, in turn bought by Eaton, and became Eaton's "BR" service panel line. "BR", Bryant, get it?
Yes, you can get a "quadplex" double-stuff which fits in two spaces and supplies two 240V circuits. This quadplex will have proper and internal handle-ties, NOT A FLIPPIN' NAIL. For instance this quadplex includes a 2-pole 40A and a 2-pole 20A. It would replace the breakers with the nail.
That breaker isn't just two twin breakers bolted together. It is purpose-built to be a quad. Internally there is a mechanism to assure both sides of the circuit trip together if either overloads (common trip). That works with or without the goofy looking handle-ties. The handle-ties are just UL listed versions of a nail; they are there to provide common maintenance shut-off. With a nail (or listed handle-tie), common trip is not guaranteed.
You cannot add any more breaker spaces to this panel. The cover has additional knockouts, but there'd be nothing under the knockouts. This is an 8-space/16 circuit panel, you are using 14 and have only 2 more possible.
After this you can add 2 more 1-pole circuits or 1 more 2-pole circuit and your panel is wedged solid. You might want to think about a bigger panel at some point. Get a panel so large that you never, ever need to use double-stuff breakers. (because increasingly, they are requiring GFCI and AFCI breakers, and those don't come in double-stuff.) You need 14 now, 24 is borderline too small, I would go at least 30, even 42.
You don't need to deprecate that panel, you could add a larger panel as a subpanel fed by a large breaker.
You could stay in "BR" panels if you really wanted to, it is still common and supported, though a bit on the cheap side. Siemens is commercial grade (so are CH and QO, but their breakers are rather expensive).