No, the springs go behind the door panels.
I don't recall exactly what the springs look like on the TR6, but if they are the same as those on my TR4 then they should stay in place if you put them on the proper way until you get the door panel on.
The springs on the TR4 are a cone shape spring and the smaller diameter end should go on the window and door stubs first, leaving the larger diameter end towards the door panel.
I just looked at my original door panels I took off my TR4 and replaced with new ones during the restoration and you can see the impression of the large diameter end of the original springs on the back side of the door panels. The purpose of the springs is to hold the door panels out against the plastic door handle bezels and handles. If the spring was on the front side of the doors you would end up with a large gap between the handle and door panel over time.
My 1933 Chevrolet uses the same exact setup.
This is for a TR4. The door panel goes between the spring and handle bezel in the graphic and you can see the proper orientation of the spring.
Vila
1933 Chevrolet
1962 Triumph TR4
1984 BMW 633 CSi