Yeah, this is SvelteKit's biggest weakness. Easy to write code that seems to work until some unusual confluence of circumstances makes it run on the client when you've always tested it on the server, or vice versa, and it breaks. I still really like it for personal projects, but I think I'd want a clearer client-server separation for anything more complex.
Is it though? Your `+page.svelte` is rendered on the server as well. Not only `.server.` is run on the server.