You've lost the plot: The [local|remote]-[open|closed] comment is making a broad claim about LLM usage in general, not limited to the hyper-narrow case of tab-grouping. I'm saying the majority of LLM-dangers are not fixed by that 4-way choice.
Even if it were solely about tab-grouping, my point still stands:
1. You're browsing some funny video site or whatever, and you're naturally expecting "stuff I'm doing now" to be all the tabs on the right.
2. A new tab opens which does not appear there, because the browser chose to move it over into your "Banking" or "Online purchases" groups, which for many users might even be scrolled off-screen.
3. An hour later you switch tasks, and return to your "Banking" or "Online Purchases". These are obviously the same tabs before that you opened from a trusted URL/bookmark, right?
4. Logged out due to inactivity? OK, you enter your username and password into... the fake phishing tab! Oops, game over.
Was the fuzzy LLM instrumental in the failure? Yes. Would having a local model with open weights protect you? No.
Depends what the side-effects can possibly be. A local+open model could still disregard-all-previous-instructions and erase your hard drive.