I beg to differ.
I was bitten three times by a small spider in my sleep at our house in Ga and it looked like the large bullseye from a tick bite. I ended up in urgent care and they gave me antibiotics. My sister was bitten by a hobo spider and some of her skin at the bite mark rotted.
Sounds like brown recluse, they can have nasty bites and sometimes the wounds won’t heal.
I keep Prid and Activated Charcoal around for spider bites. I slap Prid on any spider bite since you don’t know how bad it is at the outset.