It's like how there's this mine meta right now and the streets are full of dozens of mines at each intersection. Yes, you can shoot them, but there are just way too many and it isn't fun. I'd much rather have people make the conscious decision about which class they should play to deal with a certain situation.
The mine meta just kinda invalidates your point. Mines exist on only Engineer yet it's still a problem. And so no matter if C4 would be given to Support it wouldn't really change much. People will always pick what they find fun. In my case I'm now forced to play Recon because they are the only class with C4.
If support can heal, resupply and destroy vehicles just turns them into a super soldier where they can do everything. It leads to the spam of C4 and it makes all other classes less relevant.
Support has always been able to resupply themselves with infinite C4 similar to what they could do in both BF3, BF4, Battlefield 2042, and probably lots of other Battlefield games. Not to mention, in Battlefield 6, C4 will get put on a cool down and will slowly get resupplied anyway, which is the only reason I can still bear to play Recon.
But you are right that Support shouldn't be able to heal and resupply, and that's what began this spiral of issues with the classes being all swapped around. Making Support both heal and give ammo has resulted in a lot of disruption between the classes.