I don't know if cold water stiffens the hairs back up. I shower warm and don't let my whiskers dry between the shower and the shave. The cold water, besides being gentler on the skin, may help to stand the whiskers up better, like goose bumps do. Dunno. But there's some old wisdom that says cold water is the one true way. Old, minority-view wisdom, but there it is.
I think the hot-towel thing is part of the ritual of a full barber shave. And think about it, would you pay more to have someone lay hot towels on your face, or cold ones? Ritual isn't necessarily about what works best. It's about what the customer likes the most.