I think the server load increase from cross-instance browsing will be low. The extra load only really comes when:
Users with the fediverse-wide cookie follow a link to another instance: A tiny HTML+JS site (the embed frame) will be loaded from the content-hosting instance, in addition to the same server having to serve the content to the home instance server. This piece of HTML+JS will remain loaded as long as the client’s browser tab is open and on the domain. Even the URL can change in sync with the embed frame going to various pages thanks to the JS. As such, the load increase is constant for each session and rather small.
The one I am really worried about: Setting the fediverse-wide cookie. AFAIK, cross-domain cookies currently only work through embeds so when a user clicks “automatically log me in on each whitelisted Fediverse site”, it will need to get a response from all federated servers (feddit.de has 4250)! Many of these are single users’ personal servers and will not handle even a small ping whenever a user registers at a federated instance. We might need new web technology for cross-domain cookies or a central server just for content-less requests that do nothing but allow other domains to share a cookie, which somewhat defeats the point of the Fediverse. Please suggest another solution if you are an expert in web technology. Yes, it could be and is being solved with browser add-ons but you cannot push every new user to install one because that would just increase the barrier to entry.
Anyway, I’m quite tech-savvy but one of the first things I saw on Lemmy was “if your account is hosted on another instance, you will not be able to log in” and thought “so federation does not exist?” I hope you understand how this is discouraging: at present, federation is anything but straightforward.