Yes that's a crucial part of the issue, because many websites send their emails very slowly. So the lack of email can mean both:
"No account for this email"
"Yes, there is an account for this email but it'll become apparent much later"
And your website might send emails fast, but that won't solve the issue because as a user I don't know if your website is fast or slow, so if I'm not registered with the email I entered, I'll see no email in my inbox and I'll have to assume it's one of the slow websites and wait 10 minutes refreshing to be sure.
That's why as a user I use the sign-up form to check if I already have an account instantly, and I'd be dissatisfied by an email-based system.
I guess it's for the website admins to weigh ease of use against security, but I feel often using email worsens the U.X in a way that's not justified by the security benefits.
There's only one UX path that doesn't involve email, apart from just logging in, and that's when you do know the password, but don't know the email. In that case, you probably only have few enough email addresses to try that that isn't an issue.
In every other path, you will definitely be waiting (hopefully not long) for an email, so what does it matter?
Imagine you're an average person and you have 3 emails and 3 passwords that you typically (and repeatedly) use, in any combination. (That's pretty much every non-software person I know, and me on every site before I started using a password manager)
To login you have to try 9 combinations, but your account will be frozen after 3 wrong tries.
The log-in screen does not tell you whether you have the wrong email or the wrong password.
With the sign-up screen you can immediately know which email you used, then you only have to try the 3 passwords, and then you're in.
If you don't remember the password you can go to the forgot password screen and be confident you'll get the reset email, since you know the right address to enter.
Please read the disclaimer in the first message of this thread (which has not been edited since you replied to it).
I am describing real, widespread user behaviour, and real numerous websites. Responding "the user is dumb" or "the problem does not exist" is not how you design good U.X for your users.
"No account for this email"
"Yes, there is an account for this email but it'll become apparent much later"
And your website might send emails fast, but that won't solve the issue because as a user I don't know if your website is fast or slow, so if I'm not registered with the email I entered, I'll see no email in my inbox and I'll have to assume it's one of the slow websites and wait 10 minutes refreshing to be sure.
That's why as a user I use the sign-up form to check if I already have an account instantly, and I'd be dissatisfied by an email-based system.
I guess it's for the website admins to weigh ease of use against security, but I feel often using email worsens the U.X in a way that's not justified by the security benefits.