Each method moves dust in a different direction, and one of them pushes it deeper.
Why the grille clogs
Pockets, bags and humid air deposit lint and fine dust into the mesh. It builds up slowly, so the volume loss feels gradual rather than sudden.
A safe cleaning order
Start with the least invasive method and escalate only if needed: sound sweeps first, then a soft dry brush, then adhesive putty. Never push anything into the opening.
Try the free speaker cleaning tool and measure the result before and after.