Misty step doesn’t say the caster can bring worn or carried equipment with them. Are they intended to leave everything, including their clothes, behind?

No, the caster’s worn and carried equipment are intended to go
with them. Some teleportation effects do specify that you teleport with your gear; such specification is an example of a rule being needlessly fastidious, since no teleportation effect in the game assumes that you teleport without your clothes, just as the general movement rules don’t assume that you drop everything when you walk.