I have an old hockey injury to my left ankle. I experience both pain and weakness on the anterior of my shin, and then down over my arch -- just as you describe Machka. My physio described it as 'ankle flop' and gave me some exercises. It actually didn't do anything for me. I have to 'strap' now for every ride. Eventually I came to the perfect combo which is an elastic sock, with an ankle support overtop. Slowly the ankle has grown stronger through riding to the point where I can occasionally ride without strapping.
Another thing I've done is to move my cleats (both sides) basically as far back as I can. This takes a lot of pressure off the lever point of the ankle. The drawback of course, is that you want to have the cleat situated right under the centre of the ball of the foot for maximum power. But as a time trialist, this isn't critical; and as a rando, I'm sure it's even less important.