The handbrake gets pulled "on" by the cable. The springs are just there to release the piston when the cable is released.
I've had an on-off love affair with the rear calipers for about a decade..
The rear calipers being weak is usually caused by the piston and sliding pins being stiff.
In my opinion, the problem begins with the rear brake compensator. It can easily get lazy.. either the tensioner spring loosens off the tension over time, or the spring rusts in the extended position and then stops working as a spring. The result is that the brake bias slowly moves forwards, so that hardly any brake effort goes to the rear.
This means the rear pistons and pins don't move so much and can in turn seize or get stuck into a limited range of movement and that leads to them losing the ability to move freely, even if the whole rear brake bias is restored.
You'll see that the rear discs will develop a central band where the discs are clean.. but you get a rust band taking over the outer 1cm of the disc and another taking over the inner 1cm. Your brakes will generally fail the MOT around this point. The whole disc (bar a few mm where the pad doesn't sweep) should be clear.
So.... in terms of order of priority:
Get a new spring for the compensator if yours is stretched or rusted in an extended position. You can order it from the dealer and they're not expensive for the difference they make (and DIY fittable). Adjust the compensator so that you have maximum rear bias. The manual says hang a 5k or 10kg weight off it.. just yank on it as hard as you can and tighten it up. You can't easily put on too much rear bias.
If the compensator looks shonky, fit a new one - again not hugely expensive .. about £30.
Take off the rear calipers and make sure the sliding pins are clean (not rusted) and move freely. You can clean them up with emery cloth and grease them to buggeration (use white brake, rather than the the usual workshop grease - that tends to get baked and dry up).
Make sure the caliper pistons slide in and out easily. You need to get a brake winding tool to push in the rear pistons.. otherwise you'll damage the handbrake mechanism, which is like a ratchet.
While you're in there.. have a look at the condition of the rear pipes at the compensator. They could be twistedor kinky.. especially if the compensator has been changed before. The nuts rust to the pipe so when you undo them, theyt put a twist in the pipe which narrows down fluid flow to the compensator.
Some people replace the compensator with just a section of pipe (i.e. remove it altogether). You lose bias but presumably you gain better flow to the rear brakes and no brake problems.
My '55 is in bits for precisely this issue (and needs new subframe bushes while I'm down there) but my '45 which has all of the above works splendidly.. though I replaced the rear calipers with reconditioned ones rather than rebuild them.