The trouble with that is that it’s not like the stealth suit has a camouflage mode; people are still seeing Spider-Man shooting backs and dodging danger, requiring Ned to try to convince Betty Brant that this isn’t the real Spider-Man, but just some European knock-off.
As costume supervisor Graham Churchyard noted in Sony’s latest promotional video for Spider-Man: Far From Home, Nick Fury loans this all-black costume, which comes from the “S.H.I.E.L.D. special ops world” to Peter Parker to maintain a low profile while he’s active in Europe.