What's the technical problem with Lightning? What is making the exchanges hesitant from integrating it to complement their service? Plus are there business development people behind Lightning Labs to handle onboarding the exchanges and other institutions?
I think the main and realistic problem currently is that it isn't fully developed and hasn't had the time to prove its security.
There are a bunch of reasons different companies don't get involved, they'll be judged for everything.
On top of that it took quite a while for segwit to be adopted and this protocol requires companies to host a channel/watchtower in one location and send all their outputs out through that.
Not to mention, as soon as an exchange starts accepting the lightning network, therell be a lot of people trying to use it. And we do have more nodes than bitcoin mainnet (afaik) but each node is kinda like a miner and we probably don't have more devices signalling for the security of the ln (not that we need to get too close to that).
Exchanges work based on competition I think binance are investing in lightning research at least, so the other major large players will be doing the same...