Let's say you use 3 different sites, one for banking, one for shopping, and one for gaming. For convenience, you decide to use the same login credentials for all 3. One of them has a security breach and you are informed to change your login credentials immediately. Now all 3 are compromised instead of just one.
The traditional email process is less convenient, however it allows you to use multiple email addresses across multiple sites... thereby mitigating your potential risk.
There is no bullet-proof approach, both are equally vulnerable, however, convenience usually gives way to susceptibility in the long run. You might get compromised, you might not. It's a gamble with anything put out there on the internet.
It's far easier to dump a compromised email account and replace it with a new one than to change your Apple credentials.