Heating stainless steel mesh has been found to possibly cause the creation of dangerous chemicals (Chromium). You can't say it's safe to use. We don't know that.
This is why most people have moved to using cotton.
Yes an No. It depents on the type of SS, becaue "stainless steel" ist only a name and not qualifiying the real type.
1.4401 (X5CrNiMo17-12-2) AISI 316 ist critical.
1.4452 (X13CrMnMoN18-14-3), P2000 ist absolutly the best. (>1000 °C)
1.4006 (X12Cr13), AISI 410 >600°C good!
1.4541 (X6CrNiTi18-10), AISI 321 >900°C perfect!
1.4841 (X15CrNiSi25-21, früher X15 CrNiSi25 20) ~900°C perfect!
Which quality has the ss-pan or grill-grid, where you make your steak?
Most made mistake: Overglow of the mesh when oxidize (too long, too hot). Maximum dark Cherry-red ist perfect.
I would not want to say that cotton is absolutely secure. If it scorches it sets free carbon monoxide.
Greetings
Raidy