After the success of my crank replacement thanks to your help (SUN xcd and TA 5 vis crank differences), I did a short test ride. Everything was fine crank wise, but I had a rear-shifting issue. The lever would not stay in place with too much tension applied. I initially thought that tightening the screw would be enough but unfortunately the screw was bent. Looking for the ideal replacement screw (The previous one was the first random one I found) I stumbled upon some information on simplex shifter:
The levers have a balancing spring to reduce the likelihood of the lever slipping and as a result can be set up with less tension which makes them smoother in operation.
http://www.classiclightweights.co.uk/components/simplex-retrofriction-components.html
And then I realise that I never had such a smooth operation on my rear derailleur. I always had to tighten it a lot for the lever to stay in place. Fast forward a few turns and here it's, the broken spring. I have ridden with this broken spring since I purchased my bike from the previous owner. I will try to bend a bit of the remaining spring as per https://www.bikeforums.net/classic-vintage/1048156-simplex-retrofrictiton-shifter-broken-spring-repair.html. And try to find another suitable screw.
> Question: Do any of you understand how the Simplex spring work?
> The levers have a balancing spring to reduce the likelihood of the lever slipping
On the left, the unbroken spring from the left lever. On the right, the re-bend broken spring and the broken piece.
I will try to source either a new spring (Apparently http://www.veloclassique.com has some), replacement levers to get some springs or maybe try this washer idea.