It sounds like this bike has single pivot calipers, and that the rear brake has the cable entering from below and clamped on top - ie the cable mount part is upside down. Something like this:
Power The pictured brake has just as much mechanical advantage as if the cable entered from above and got pinch-clamped on the lower arm, (probably like the front caliper)
This setup looses some power due to the additional frictional losses involved with that large sweeping bend, and a little to the additional length.
If you have new modern inner cable with a coating, and outer housing with teflon liner, then you're as well-set up as you can reasonably get.
Centering Cable-operated brakes all use springs to pull the pads off the brake track. On an older bike, its totally possible for one side of a spring to lose tension at a different rate to the other side of the spring. Some calipers have little blocks that push the arm out, adding tension to try and keep them about the same. You can also try flipping the spring over.
Pads Traditional brake blocks were plain rubber. Modern brake pads tend to have much more technology in them, and even a simple one-for-one replacement with good new ones is an upgrade to braking. This may mean buying brake pads that have a permanent shoe and a consumable insert rather than the one-piece items we used to have (as pictured)
Wheels Does it have steel rims? If so, one excellent upgrade is Aluminium rims. The braking track has much more bite and simply works better. Downside is a rim will wear out faster than a steel rim, but that's still many ten-thousands of kilometres.
You might also find that modern rims are 622mm, whereas your older bike might have 630mm or 635mm rims. This will give a much wider selection of modern tyres, but the brakes have to reach further and therefore loose mechanical advantage.
Personally I'd go for aluminium rims, with dual-pivot calipers, and kool-stop brand brake pads. If you can only replace one rim, make it the front wheel.
For calipers, there are "long reach" which will help you overcome the slightly smaller diameter of modern wheels, for example Tektro R559
Nobody seems to sell a cable-direction swapper adapter, so you might have to steal parts off the old caliper to make a new one work, or possibly put up with an S bend in the cable outer's path.
Another option is to use a brake "noodle" from a V brake to help turn the wire's path by 90 or 135 degrees, but this may need some finangling to work well.
Expanding on a noodle:
The noodle is the silver tube seen here on a V brake that "turns" the pull of the inner cable by 90 or 135 degrees without the longer length needed by outer for the same turn.
I've used noodles at other points for the same purpose - pictured here is a folder with noodles allowing the gear and brake cables to take a relatively sharp turn. These happen to be flexible noodles but rigid ones would work the same here.
I don't own a mixte frame, but imagine the green line is the missing tube and is the highest path your brake cable can follow. The pink line is the cable outer's path, and I've clamped some parts to the rim as a mock-up.
Another shot of the mockup. Your challenge is to clamp the inner cable wire at the top arm, and put the outer cable-endstop in the lower arm. This may be possible with machine tools and fabrication, or you may be able to move parts from the existing caliper to a new one.
Refurb the current brake and be aware that it's kinda rubbish. Upgrade the front brake to a dual pivot and learn to brake more with that. If you can find a replacement fork with disk brake mounts, then a new front wheel with a brake rotor might be an option though this is getting pricy.
Check the middle tubes and the chainstay area for another brake mounting hole. You might be able to relocate the rear brake caliper around the wheel a bit, permitting a straight-line entry for the cable.
Perhaps a single noodle will allow you to mount a normal caliper, and still get the cable routed sufficiently well.
There exist products like the Problem Solver's Cross Clamp which can turn a bare cable by any amount up to 180 degrees.
although these have no way to terminate the outer housing.
Also there are "travel adapters which work like a V brake noodle but use a pulley to change direction.
These have an outer-cable stop and might work similarly to a noodle but again with more angular change. https://problemsolversbike.com/products/brakes/travel_agents_-_6416
Apparently there exist old Weinmann calipers which are convertible from conventional top entry to bottom entry. However they're vintage, likely expensive, and probably still rubbish at braking.
I suggest a Dual pivot caliper for the front, and refurb the existing caliper as much as possible for the rear. It'll never be great, but two brakes makes it legal. Anything else has higher costs.