Yes, people will have to wait and download the whole blockchain, hopefully the bandwith will increase by that time.
Current bandwidth is more than sufficient to download the entire blockchain within a few hours at max. Problem is that there exist a ton of bottlenecks which can't be eliminated that easily. I don't think there is a particularly huge issue with an increase in the rate of transactions in the future; assuming transactions can take place off-chain and we can sustain an increase in the efficiency of the various components. I think Moore's law isn't that relevant anymore. We're already reaching 7nm nodes.
I would be more concerned about longer perspective, like 100 or 200 years, if our storage and bandwith technology will stagnate, eventually running a node could become a big problem, and the network would be slowly becoming more decentralized. Perhaps a blocksize reduction would be needed at some point in the future to combat this. But this is a problem for future generations.
Reducing block size is counterintuitive. We need bigger block size to support LN, even if we were to scale to our closest payment system competitor. It's quite evident that on-chain capacity isn't ideal right now, outright blocksize increase cannot be the only solution.