When temporary email is the wrong tool
Temporary public email is a narrow tool. The useful cases are small: a template check, a code for a demo with no real data, a look at a subject line. The following situations should stay on a mailbox that has its own login.
Anything that identifies a person
Identity documents, medical notes, school records, employment files, and home addresses do not belong in a public inbox. Skipping a signup is not a reason to publish them. If a test message would be awkward on a noticeboard, do not send it here.
Money and access
Banks, card issuers, payroll, tax, stores of value, and any password reset are out of scope. A public inbox cannot be the recovery method for a valuable account. A single "just this once" reset is enough for someone else to request the next one. Read do not use a public inbox for passwords if that tradeoff is still unclear.
Mail you must answer
Support conversations, job applications, and vendor threads need a reply and a record. This site has no send button and no retention promise. A message you may need next quarter should live in a mailbox that can still find it next quarter.
A domain the other site does not allow
Some products reject domains that are known for disposable addresses. If a form will not accept the address, respect that. The form owner is allowed to require a mailbox that keeps a history and a login. This site will not disguise the domain to get past that choice.
Use a mailbox you log in to for anything in the list above. Use this site when the message is a test artifact and a stranger reading it would not create a real harm. What a public inbox is is the short version of that boundary.