You could ask the opposite question as well:

"Please list businesses whose success depends on frothy, bullish market conditions"

Most companies can be successful even in down markets. See basically all S&P 500 companies. Yes their stock price might be down and they may be laying people off to save money, but those are all legitimate and successful businesses by most measures.

Theoretically, localbitcoins should be very insensitive to the bitcoin markets, because for them it doesn't matter much what the price is -- they just facilitate transactions.

If transactions are still happening, regardless of the BTC price, there has to be a way to make money from facilitating them. And given that BTC has been running with full blocks for years, the transactions should be there.

Plus, it's existed for a long time, so at this point the service has been built. If business is slow then it can just coast on what already exists.

Of course it could indicate that nobody actually uses BTC that way anymore, and the current transactions are of a different nature than localbitcoins was made for.

Tobacco companies. No matter what happens, they just seem to be making more money.