Many of them do, at any rate.

Edinburgh Folk Club never used to do this and, as I recall, proved very successful during the tourist season. It even put on its own Fringe Shows.
Changes in venues made it less practical to continue especially after moving to the Pleasance which is always hired by Fringe companies during that time.

However, I notice that most clubs, especially in Scotland, seem to close their doors at this time of year.

I can understand that the organisers may wish a break and some of the regulars may be on holiday too but I'd have thought passing trade especially in more touristy areas woud more than make up for this. Also, I'd be more inclined to leave the house on a Summer night too.