As I understand it today, I can view an ad on a random site and have a cookie with my unique ID in it saved. If I view other network ads, they know it's me and update my profile. Then when I check out, the site I'm buying from reads this cookie and reports back that it worked based on impressions.

AFAIK this won't be easy to replace if a random eCommerce store is blocked from reading Facebook et al cookies, it won't be able to report back reliably. You could do something like submit their email address or other data to Facebook and see if you get a hit, but that's probably illegal in many places without permission. And in most cases, the cookie from the ads will be blocked as it's third party (but would be able to be written for ads shown on-platform, which doesn't really matter since they can track server side anyway).

But clicks are very easy, you just tie a unique ID to the ad URL and have the landing page and checkout page track that. No cookies needed. It can report back in real time or later to update stats.