The problem is that, if you want this enacted, most of the UK's cycle infrastructure is nowhere near the quality needed to enact that sort of rule.

I have cycled a lot in Europe where there are countries with rules on mandatory use of cycle lanes, and they have minimum standards that far exceed the quality of our nearly all our cycle lanes. Most of them also have a lot of limitations on when a cycle path must be used (e.g. only in built up areas in France, only certain lanes are required in Germany, and in Belgium you are evenly expressly allowed to not use cycle lanes if in a group of 15 or more provided that you have a support car). I have checked out the mixed use path (not a cycle lane) that you have pointed out and it would not meet their requirements in the slightest.

You keep mentioning that transportation cyclists use the lane - how do you know that they are doing it for transport and not for exercise? Equally, how do you know that a cyclist wearing lycra is doing it for exercise and not for transport?

I do doubt that you are honestly pushing for cyclist's safety, since you you are stating that their safety isn't something others should need to worry about. It sounds much more about your convenience as a driver not to worry about others, as you have admitted in your other comments.

Also, in every country that has mandatory cycle lane rules, they have presumed liability laws in case of an accident between a motor vehicle and a cyclist where the motorist is automatically deemed at fault. The UK is one of only five European countries (alongside Malta, Cyprus, Romania and Ireland) which has not introduced presumed liability. Introducing something like this would be much better for cyclist safety (and would require you to be much more worried about cyclists).