Agreed with many recent posters. Bottom line is your son has to have the skill set to help them win games, that is priority #1. "Unofficial visit" invitations are usually extended because a coach has already seen a player demonstrate this.
To be fair, what you are talking about in OP are not technically UVs, but Unsolicited Visits, or visits where you initiate the contact and hope for an audience, and that's fine. But at best, all that can come out of it is the coach may think, "Kid looks good on video, his posted metrics on ABC Scouting Service website look good, seems like a nice kid, respects his parents, dad is huge, mom is a nice lady, I will try to see him play in Atlanta next summer." And if a cold call email barrage, gets you even one of these audiences, and the coach goes out of his way to see your son play at some point? Well, then it's worth it.
But understand, it is highly unconventional and sort of putting carriage before the horse. During son's process, any time he could walk a campus, speak with a coach, or even a HC, even if the guy was on his way to a funeral, that's worth it. It will get your son acclimated to those encounters and get him that much more comfortable for when it does count and the more campuses he gets on the more informed a college decision he will make.
I say good luck, but I also say maximize your strategy and dollars on the trip.