Is removing Bootstrap from a five page site worth the effort?
Usually yes at that size. Grid and flexbox cover the layout, custom properties cover the theming, and you stop shipping 200 KB to style five pages. The part people underestimate is the component behaviour, not the styles: modals and dropdowns have to be rewritten.
Switching to dark theme flashes white on load. How do people avoid it?
Set the theme before first paint with a tiny inline script in the head that reads the stored preference and puts an attribute on the root element. Anything loaded as an external file arrives too late, which is exactly what the flash is showing you.
At what point does a static site need a backend?
When you need to keep a secret or write shared state. Everything else, including search over your own content, can be done in the browser with a prebuilt index, and it stays working long after any server you would have written stops being maintained.