What do you use Live View for? I keep a Safari window open [Safari version 18.4 under macOS 14.7.5] and have only used Live View for the Orders tab. Safari has eBay’s website open, and updates my active listings seconds after GS has listed them. I also have a tab open with my items for sale listed by time listed most recently. I check that a couple times a day to make sure I didn’t make some dumb typo in the title or price something way off.
EDIT: I’ve only had the checking your browser thing come up when I am on a VPN. So maybe the IP range you’re using has been flagged by eBay as having some scammy issues?