Plumbing question related to ice maker drain

I have purchased a small commercial ice maker for a camp and the instructions say to drain it into an open drain with at least a 1” gap and no P-trap. It is going next to a utility sink and I’ve prepped the plumbing to drain into the 2” pvc line p trapped that then runs parallel for 2-3 ft before going under the slab.

Any recommendations on how to run this? It’s a gravity drain and will be high enough to gravity drain into the 2” line. I think I’m going to run the 1/2” drain into the 2” pipe through a T reducer ( 2” down and up, 1/2” perpindicukar into it) then have a 2” pipe vent up to allow for the required 1” venting. Any reasons why this wouldn’t work? I don’t understand why a Ptrap wouldn’t work outside of it spills out more water then the ptrap can get rid of but that would be almost impossible as the water line in is 1/4” and the lines out are 1/2” to 2” pipe.