What you have appears to be what is listed as NRP Hinge (non-removeable pin hinge). How do you intended to fix the door not staying open with the hinge pin removed?

@millworkman they suggested that I bend the hinge pin slightly and pop it back in. The bend would add resistance and prevent the door from closing automatically.

Could a missing hinge pin on the jamb cause this issue?

Also, i am surprised they used non removable hinge pin as it is bathroom door.

Same type of hinges that are in my house. Pins are removable. Find a nail that will fit in the hole & use a hammer to tap the pin up enough to be able to remove it. (Or a really skinny nail set or an ice pick...)

Putting a bend on the hinge pin has worked for me. Worked for a couple of years--then had to redo it. (Bent only one pin.)

Awesome! Thank you @ci_lantro.

This door has two hinges. Did you bend the top hinge or bottom one?

Mine has 3 hinges. Honestly don't remember but almost sure it was either the top or middle one because those are easier to get to vs tapping out the pin in the bottom hinge.