I tried to mark all disconnected edges with sharp edges and merge them by distance, which didn't improve the problem. Does anyone know a good fix to use the edge split mod without loosing the quality and without disconnecting the edges? Would be great, because I used this mod on almost 70% of the whole character. Changing them all by hand would be terrible.
Yes. That's what an edge split modifier does. If you don't want to disconnect the edges, don't use an edge split modifier. If you're trying to use an edge split modifier to create hard edges, there are other techniques (mark sharp, autosmooth, which it appears you're already using); if you need an edge split modifier for an export somewhere, don't apply it until after you weight, if it's screwing you up.