Shouldn't be. The relay that is used by NWC is part of the connection string itself and is therefore completely separate from your regular relay list.
Discussion
On #Amethyst, I couldn't zap using #Coinos. So while trying everything, I added Coinos relays in Amethyst and then I was able to zap. The issue could be somewhere else. But that's what worked for me. And this is why I asked. So should I remove Coinos relays and try again?
nostr:nprofile1qqstwf6d9r37nqalwgxmfd9p9gclt3l0yc3jp5zuyhkfqjy6extz3jcpz9mhxue69uhkummnw3ezuamfdejj7qgcwaehxw309anxjmr5v4ezumn0wd68ytnhd9hx2tcprdmhxue69uhhyetvv9ujucnjd9nksarzdak8gtnwv46z7273e59 okay, it worked now. I removed #Coinos relays from #Amethyst and tried Zapping. This time it worked. But earlier, as I was, when Zapping wasn't working, I added Coinos relays and then it worked. Weird! But at least it worked now without the relay. Thanks.
Glad to hear it is working now! Not sure what the deal was before that it wouldn't work without the CoinOS relay in your regular relay list. This is the only place you need it to show:

Yeah, #Coinos relay showed up here. But I'm not particularly sure whether it was here when zapping wasn't working. I'll keep testing it. Thank you for the tips. Really appreciate it.