Can a website operator prove I consented to their terms if I block their consent popup?

If you continue to use their website than that is a you problem. It is no different than actively ignoring the signage at the local kroger saying "no guns allowed"

If I block consent notices how would I possibly know there was a consent notice governing continued use and how would a company know I never actually saw the consent notice to begin with?

I also don’t consent to having billboards all around me or ads literally mailed to me in the post.

Which is a very different mess with very different laws governing it. That said? You would be shocked how easy it is to complain about a billboard ad and get it to go away.

It's the same mess. A company makes an ad and partners with another company to distribute that ad. That distributor then partners with several vendors to show that ad. In exactly 0 cases was the recipient of the ad asked for consent. In one case the recipient of that ad has an option to not see it--heaven forbid they actually exercise that option.