3-way switches with receptacles
Based on your details on what you did with the replacement outlet, I'd say you didn't remove the metal tab on the outlet connecting the two brass, power, screws. Doing this allows the two outlets to be split between always hot and switched hot.
The one wire you &quot;punched&quot; into the backstab should be pigtailed so you don't use the backstab. they are very prone to failure.