A horse shouldn’t need boots after trimming, even if therapeutic trimming needs to be done. It shouldn’t be painful; at least not in fairly normal cases (in comparison to rescue situations where there is major work to be done.)
If the trim is the cause of the inflammation/soreness, the farrier is trying to change things too quickly. Crippling the horse is not necessary to correct a bad trim.
That said, boots aren’t a bad thing to have - but they should be necessary only after a trim. If the horse needs boots, put boots on it.