Am I missing something, or is the stock on-screen keyboard just completely unusable?

[worldly rune]

I can either have it set to automatically show up which doesn't work on browsers making it basically useless, or I can have it permanently take up a quarter of my screen, which is unacceptable especially considering it's already quite small to begin with.

[drowsy lance]

While you are looking for a more elegant solution, a workaround: Go to System Settings > Keyboard > Shortcuts and set a hotkey to toggle the on-screen keyboard.