I would assume they're motivated to cut down on their own tx fees.
BitMEX does not pay the transaction fees. I guess that's part of the reason they still use their tech from 2014 for withdrawals. Quoting from my article:
The miner fees for withdrawals are not paid by BitMex. They are deducted from the withdrawing users. The transaction feerate does not depend on the input count nor the transaction size. Users choose a miner fee when withdrawing. However, they do not know how many inputs the final withdrawal transaction will spend and thus can not reason about the transaction size and required feerate. All observed transaction fees are a multiple of 10 000 satoshi, which is the minimum step size the withdrawal frontend allows. 10 000 satoshi is the smallest, and most commonly (44%) observed withdrawal fee. It is followed by 100 000 satoshi (30%), 20 000 satoshi (17.5%), and 50 000 satoshi (3%).
I could imagine the same thing existing for bitcoin-- but the security considerations would probably make it a harder sell.
I thought about creating a service similar to what you describe. Reasons I didn't are that exchanges/other services often don't pay the transaction fees themselves, I don't have enough
exchange-security experience to not mess anything up (at least I'm not confident and insured enough) and I'd be (at least partially) competing with the Bitcoin Optech guys.