A tyre that needs air every few days usually has something in the tread, a leaking valve, or a poor seal where the tyre meets the rim, often from corrosion on an alloy wheel. A tyre pressure warning light that keeps coming back after the tyres have been pumped up is a common sign.
Tyre sealant from an emergency kit is meant to get the car to a safe place, not to fix the tyre. The sealant has to be cleaned out before the inside can be checked, and some sealants make a repair impractical, so say on the call if a can has been used.
Run-flat tyres let the car keep moving for a limited distance after a puncture, but that driving can damage the tyre from inside. Some run-flat makers advise against repairing their tyres at all, so the fitter follows the maker's guidance for your tyre.