You are a good mum. You care and you want to change things that you feel aren't working well for him. Sign of a good mum. I agree that you really must let your DS do a couple of after school activities - Beavers and tag rugby or football. Beavers will be one weeknight and the sport one weekend morning. Decide this is manageable. Just decide that it is. If he gets any good at sport, he will have automatic mates so give him the chance to develop those skills.
If you have faith or are curious about it, you could consider going to church. Big family friendly churches have age appropriate groups that leave the service and play games for most of the morning. They are very inclusive. He would build up a group of children who he is familiar with over the weeks, and they end up being close because they grew up together. Also with the younger groups, they focus so much on Jesus loving him, and being a friend to him, it could be quite reassuring.
As to playdates - tidy and clean the hallway, living room, his bedroom and the bathroom. Shut the other doors. Use Flylady 15 minute Room Rescue techniques to achieve this without overwhelm. Put out a couple of toys they might want to play with - lego or train set or garage or playmobile, and set up a simple video game or good fun age appropriate film.
Supply a very simple tea of pizza with a few veg sticks and either banana custard or a scoop of ice cream for dessert.
Pretty much leave them to it unless they seem bored or are fighting, I'd do: 20-30 minutes playing, 20-30 minutes have some tea, then 30mins to an hour on a game or some cartoons.
You can also invite children on playdates outside the house - invite a child to come swimming or bowling or to a city farm, then take them for a slice of pizza or a burger afterwards.
Don't tell him unless it is already arranged as you don't want to raise his hopes. And try to reassure him he will make friends. Showing your own anxiety about it will increase his. DS2 had no friends for a long while. Now he has so many he can't keep up with them all. It comes in waves. Just keep his confidence up.
Good luck to you in making some changes to help him make friends. It's a really lovely, good thing to do.