How to wire 3 hots / 3 neutrals into this new TR outlet with 2 screws?
Your two options here are to pigtail the hots and neutrals, or get a new "backwire" outlet.  A backwire outlet has small wire clamps on the sides of the outlet so the wire can be inserted straight in rather than looping around the screw.  This type of outlet might be sold as "heavy duty", but they are great for handling up to 4 conductors on each side.  The ground will still need to be pigtailed, but that's usually not a big deal.



Example outlet

If you look closely at that picture, you can see 4 holes in the back on each side.  Wire inserts straight into the holes and the screws on the side tighten the wire in clamps.  They are very similar to your old push-connect outlet, but they clamp tight on the wire.

Backwire outlets are great for smaller boxes because you don't need to worry about a couple 4-wire bundles of 12GA wire getting pushed into the box along with the outlet.

A pigtail is used where there are too many converging wires to be accommodated by a fixture, such as you describe here.  The converging wires, and another short length of similar wire (the pigtail), are twisted and secured by a wire nut.  This leaves you with a single lead to attach to the fixture.  (Added for completeness)