Pros/Cons of buying/holding bitcoin through 3rd party custodian

If only for investment purposes, what are the pros/cons of purchasing and holding bitcoin through say, fidelity crypto, versus purchasing through an exchange and then transferring to a cold wallet.

Again, my purpose is not to trade crypto or purchase other items with crypto, only to hodl and watch appreciate in value.

I’ve heard people say it’s not truly yours until it’s in cold storage but shouldn’t Fidelity be responsible for any theft/hacks since they’re assuming responsibility as custodian?

My take is having someone take the responsibility of owning and managing “my” bitcoin is well worth the spread of 1% when buying/selling so as not to have to deal with hardware, seed phrases, passwords, etc.

I’m just wondering if there’s any other benefit to self managing bitcoin I’m not thing of.