> I'm guessing it means that your use of the website is not contingent on you accepting

I don't think it says that at all. Because "accepting" is the right word for this interpretation, as you point out. "Approval" is a different thing altogether. You can accept something without approving of it -- that's the main message in the Serenity Prayer and hundreds of self-help books that try to reframe that message, maybe to help it sink in, maybe just to grift a little.

If it was literally spelled out as "Your access is not conditioned on your approval" that could almost be taken as a threat -- you will access this whether you want to or not.

> For instance, your access of the website is not contingent on the website operator approving said access.

To me, this is clearly what it says. "(Your) access is not conditioned on (our) approval."

But, of course, since you read it differently, I have to agree that perhaps it's not as clear as I thought.

However...

Contracts and agreements, if ambiguous, are always interpreted in a light most favorable to the party who didn't draft them.

So, absolute worst case (for the website owner), if we combine your reading and mine, it reads "Your access is not conditioned on either your approval of these terms, or our approval of you."

Somehow, I think the author is OK with this.