it defenitly does, although I dont know exactly where the bottlenecks are. e.g. the three network has capacity problems.
Generally for a start masts still need a backhaul feeding them, so that backhaul itself can get congested, a bit like backhaul to a telephone exchange. I think also the wireless connectivity itself still has a capacity limit, from what I understand higher frequencies have more capacity but also shorter range. This is one reason why EE is high performing for mobile network standards as they got a big chunk of the higher frequency stuff.
For the last question I think it will at some point.
Look at whats happened in home networks, all the focus has been on wireless connectivity, so ethernet has stagnated at gigabit on home routers, but now AC wireless tech can surpass that. There is also already work on a successor for AC tech yet 2.5gbit or 10gbit ports on routers seem miles away. 5G will have 6G after it and so on for mobile networks, whilst for wired broadband I dont know of anything in development that would surpass FTTP.