[upofadown]

If it is an actual legitimate interest then you would likely be expected to contact the site out of band to object to the use of your data. Depending on the technical details you might not be able to continue using the site after a successful objection. In some cases the site might be able to reject your request.

The cookie banner thing is intended to allow the user to explicitly provide consent, should they for some reason wish to do so.

[Rygian]

The cookie banners are routinely used to object to "legitimate interest" uses and the corresponding sites continue to work normally, not sure what your alternate understanding is based on.

[upofadown]

The cookie banners are for initial consent. You just consent to less stuff sometimes.

A website might claim some sort of legitimate interest for the initial collection of data but might not think that they can claim that for the retention of data I suppose. That would seem kind of dodgy to me...

Just because a website claims something doesn't mean it is valid. There isn't a lot that falls under legitimate interest for a website.