Self-custody means the assets are controlled by a key you hold. No platform can freeze the balance, lend it out, or fail while holding it, which is the property the phrase "not your keys, not your coins" points at.
The case for it is written in bankruptcy filings. Customers of Mt. Gox, Celsius and FTX discovered that a balance shown in an app is a claim on a company, and that such claims are settled years later at a fraction of their value.
The responsibility does not transfer to anyone else. Lost keys, a phishing signature or a compromised recovery phrase are unrecoverable and uninsured, which is a genuine trade-off rather than an obviously correct choice for every holder.