MadDoggy, just continue strapless
Usually I would advocate straps, but not in your case.
The reason taxi'ing is great, is ONLY when you have a big board or flat water.
Then you can learn gradually to control things, and you can avoid the crazy "foil flies up in the air" which can give dangerous situations.
At least it helps you dont have straps - so feet wont get trapped.
You say you have a good feel balancing the board strapless, great (not many has, when learning)
As the foot position is extremely sensitive, a few cm too far forward or rear will make it impossible to fly, or it will rocket out of the water.
Would recommend you, if not already present, make some kind of marking on your board, where you think your front foot should be.
It does not need to be accurate, and better a tad too forward than rearward, much easier to spot now.
Now you can put your front foot much faster in position, and even when waterstarted you have a few seconds to move your foot.
Having such a reference to aim for, will help immensely, even for experienced.
It doesnt have to be precise, as you will very fast learn where it works best for you, putting your front foot relative to the marking
You could use a strap of some kind to obtain the same, but with your progress and conditions, I wouldnt do it.
And to your question "Does straps help control once foiling?"
No, not at all - they help for positioning the board when waterstarting, and can help when touching down on the water - but when riding, no help whatsoever for balance
Take the time it takes to learn foiling, it is REALLY difficult because the foil is crazy sensitive to foot pressure, but it will come.
Peter