6800 products isn't a lot.
If they're heavily laden with attributes, that does certainly create additional burden, but shouldn't seriously impact displaying a product listing.
A REPAIR or OPTIMIZE will cause a temporary lock on the tables while it does the job, and customers will just experience a brief hang if they click while that lock is in place until the lock is released. If your store had 10x that many products then I'd suggest waiting until a slow time to do it.

The slow-query-log might be needed for you to see the details of the query.
Or a GUI tool like Navicat or SQLYog which can extract more information from the database than phpMyAdmin does in things like SHOW PROCESSES and other things.