I have never been a fan of trying to push a mute strip / temperament strip down past and to below the damper felts, exactly for this reason. Operator error.That being said, I prefer the center-stitched damper felts because they are less prone to damage, inside and out of the piano. Also, I think they are less noisy, plus they allow half-pedalling.
If you get the contract to fix this mess and replace the damper felts, you may want to look at the center stitched kind. They are available for uprights as well as grands.
I was called once to a fairly new Kawai upright and the customer told me all the strings are ringing. I went. Mice had set up shop and used all the dampers for a cozy camp! Under warranty, so all good.