she was probably tired from walking to Donwell (no carriage since horse is lame) and so she is conveniently available to babysit Mr. Woodhouse. I doubt, now that she is married, that either Emma or Mr. Knightley would have expected Mrs. Weston to have the same responsibility she had towards Mr. Woodhouse as Miss Taylor.

As to the idea of Frank being responsible for the lame horse, I am not so sure, though it sounds plausible. Highbury is such a small place that most people (including Frank) seem to take their exercise by walking than riding. The only time Frank could have used the horse was to dash off to London and back in his previous visit.