Financial institutions have to log a lot just to comply with regulations, including every user activity and every money flow. On an exchange that does billions of operation per seconds, often with bots, that's a lot.
Yes but audit requirements doesn't mean you need to be able to search everything very fast.
Binance might not have a 24/7 constant load, there might be plenty of time to compact and write audit data away at lower load while leveraging existing infrastructure.
Or extracting audit logging into binary format like protobuff and writing it away highly optimized.
Binance boss just came out of prison, so he is not above the law.
And yes, they likely have this problem, because crypto is less regulated and full of geeks, so they have way more automation compared to traditional finance, at least proportionally to its size. You have bots on top of bots (literally, like telegram bots that will send orders to other bots copy-trading other bots).
In fact, there is even an old altcoin dedicated to automation: kryll (https://www.kryll.io/). They have a full no-code UI to create a trading bot with complex strategies that is pretty well done, from a purely technical perspective. They plug into many exchanges, including Binance.
Also, because it's less regulated, copy trading/referral/stacking is the Wild West, and they generate a lot of operations and fees.