This assumes all requests are made by the owner of the e-mail address, with sufficient understanding to give consent vs. e.g. an e-mail provider that fails to understand the privacy concerns.

Even then, as pointed out by others, this does not preclude the user from withdrawing consent to continued use of the information.

It may or may not become a legal problem. But they really need to consider the privacy implications and have lawyers that actually know the relevant national laws throughout the EU member states to evaluate it.

in EU, you can say at any point to a company 'delete every piece of personal information about me that you have', and email is considered an personal information. Even if you gave consent, you can withdraw consent at any point in time.

So, basically, I can tell my bank/credit agency/hospital "please forget that I got a mortgage/defaulted/tested positive for HIV"? Doesn't that sound rather risky?

It's not like that. You can demand to be informed about when the data is processed and for what purposes, get a copy of it, request corrections and object to processing in certain circumstances.

In practice, no, you can't demand to be wiped out from the internal database if you creditor. You probably can prevent them from publishing a debtors' list with your name, though.

It's not that simple (in the UK at least), you only have to stop processing information if "unwarranted and substantial damage or distress" is being caused.

Additionally, all stop processing / deletion regulations require a court order, which will be subject to all the usual requirements of justice.

It only takes a single successful demand before an append-only log because untenable.

There are of course exceptions ;)

Yes, but if users later can't remove their information because the list is append only then that may not be legal in the EU.