In Skyrim, is there a way to not automatically skip the loading/tips screen, but require a button press?

I have an SSD, so the load screens last for a fraction of a second. I'd like to be able to read the hints and view the items displayed, and continue when something is pressed.

If you mean the screens that display some NPC or item model (which you can rotate via your mouse) when swapping world spaces (e.g. between a cave and the outside world, for example), then no, there is no way within the game or the file settings to make loading screens wait until button press before continuing to the next one or on to the playable game.

However, some players on Reddit and other forums have suggested that, because Skyrim's loading rate allegedly depends on your framerate, you can install a heavy graphics mod like ENB to slow down the loading rate of the game. Coincidentally, ENB allegedly also has a keyboard shortcut to toggle off and on, which could theoretically be used to skip loading of ENB settings/textures once you've finished reading a screen... but that is only one small part of your request.

I have never tried ENB so I don't know whether that is still (or ever actually was) a feature/effect of the mod.

See https://www.reddit.com/r/skyrimmods/comments/2tydii/pause_loading_screens/co9pz9t/ for info related to this claim.