20 months is far to young to move baby out of a cot imo, unless you cosleep. So defo keep the cot.
If he's crying, I wouldn't leave the room. But I would carry on doing what you are - having him lie back down every time he gets up.
I'd suggest instead of you lying him down, that you start working on instruction following and get him to lie down himself without your help.
Play some instruction following games during the day. Practice instruction following constantly, all day. Things like:
- put this in the bin
- can you bring me the ball?
- take your plate into the kitchen
- take this to Daddy for me please
- can you touch your nose/elbow/ears etc?
- I want you to sit here
- can you stand up please
- "Mummy says.." games, like Simon Says but with all if them being an instruction. Roll over, stand on one foot, lie down, stand up
Generally give instructions for DC to do things independantly whenever possible. Tons of praise every time he follows znd instruction you give. Make it so he wants to follow the instruction.
Then at bedtime, stop lying him down. Stop completely, don't do it at all at any time. Put him in the cot standing up, tap the mattress, say "lie down" and wait. Big praise, eye contact and close physical contact (ie you leaning into the cot) when he follows the instruction. Stay close by while he lies down and settles.
Then every time he goes for sitting up or getting up, tap the mattress again and repeatedly just keep getting him to lie himself back down.