I might actually recommend it for anything except financials. Financials are very important to have access to, and sometimes your email address is the only way to access your account.

~~I want something very stable as my email address for such cases, because I don't want to risk my email alias provider shutting down, or my self hosted setup to randomly die. Or even some weird "security measure" where the bank decides that any unknown domain is no longer ok.~~

In such cases, I want my email address to be reliable. Email alias providers shut down from time to time, and I don't trust my skills to set up a reliable self hosted option. If my setup breaks after an upgrade, I might be out of luck till I can change my email with the banks.

Banks also have their own policies, and sometimes they make questionable decisions in the name of "security". Companies already block certain alias domains, and there are stories out there of people being locked out of their online accounts after a domain was blocked. Banks are meant to be trustworthy so they may take an even harder stance on it. They may decide to restrict the common alias domains (ex. citing that spammers use them), or even restrict personal accounts to the big email providers (ex. Gmail).

For example, some bank apps don't work on phones with a custom OS (grapheneOS). Singapore went even further with:

Local banks DBS and UOB are rolling out new anti-scam security measures that include restricting customers from accessing the banks' digital services on their mobile phones if apps from unverified app stores – also known as sideloaded apps – are detected

Ultimately it's a lot of risk for a very small reward. If you use a different password for everything (you should), then someone knowing your email still won't be able to do much. The bank itself would have way more information about you, so it's not like you're protecting yourself from the bank with a custom email.

Although I agree with you, I don't think that's what OP was asking about based on this part:

I'm just thinking that if a hacker got access to one email they'd have all account information?

It seems they are asking if an separate email account for each service would be beneficial. My opinion is it would limit the attack if an email account was hacked, but definitely not worth the hassle. Email aliasing (like the comment above me says) gives you some of the benefits without needing to juggle multiple accounts.