Thing about it was Mom had more than one set of teething rings. Several in fact in the fridge. We’d have one in our mouth then when that melted, put it back to re-freeze and get another. Never heard of using alcohol to do the job, just cryogenics.
I had a hard teething ring but I hated it. My (and my husband’s) pediatrician was an old (then) country doctor who often suggested things like this. My grandmother did it with her parents too.
Not sure how old you are but teething rings are fairly new.