"Would the French drain not work properly because of we filled it up?"

If you fill a French Drain with concrete it will never work again as the place where the water would flow is now solid concrete. So if you needed a French Drain in the first place this to me sounds like an assanine work around.

Then re-test for radon and my issue might be solved without needing a radon system.

And when the water comes back?

Install the radon system and re-test. Then you'll know if it's time to use an alternative drain system, typically from the exterior.

No, you should not fill the channel because it sounds like it was done to mitigate a water issue. Pictures would helpful. Pictures of drain and sump. What year is house? Full basement?

Filling the gap depends why the interior drain system was installed and what they installed.

Typically, a homeowner sees water in a basement and calls an "expert basement waterproofing company." These companies are 1 trick ponies that apply their product to any problem no matter what the cause (sorry for the rant). So if the gap has some plastic material sticking though, it is probably something like one of these two systems:

The main purpose of these systems is when you have water coming through the wall, that gap catches water running down the wall and directs it into the pipe. So filling that gap would be problematic.

Or they could have installed just the pipe like the left picture and just left the gravel be exposed instead of filling back the concrete slab.

However, if the problem was a high water table working it's way up from below the slab, or the walls are intact and water is seeping in at the footing joint, then that gap really serves no purpose, and can be filled.

A typical radon system pipes to below the slab and then pulls air out of the sub-slab, and depends on the existence of porous gravel under the slab to allow air to move towards the pipe. You may not get adequate results leaving exposed gravel at the drain. Some photos of the drain and a description of the system would help.

Where was your radon test performed, in the basement or upstairs, and how much beyond 4pCl/L?

A house from 1930 is probably very leaky. If a low reading upstairs, the radon may be dissipated enough just with the natural basement airiness (gaps in windows, poor windows, gaps at rim joist area, etc). If so, and you don't really have living/sleeping space in the basement, you can have a little patience in addressing the radon until you live there a while and get to know your house - like understanding if the water problem is the walls too, when the sump pump runs through the seasons, and what's going on outside with water control. This could determine how you approach finishing the basement someday as well.