Enclosing my front courtyard, how to wire video doorbells

I’m enclosing my front courtyard, so no one will be able to ring the existing push button doorbell by my front door.
My existing doorbell rings my mechanical chimes upstairs and down. It is wired with a standard 16v 10 amp transformer.
I plan to install 2 Ring video doorbells in the columns supporting the new gates which provide entry into the courtyard. I had my contractor run wiring from the columns to the transformer, adding that wiring to the existing wiring from the front doorbell. So the transformer had 2 wires on each post; one set from the existing front doorbell and one set from the columns.
But when I touched both wires in the column together to test if it would ring my existing mechanical chimes, nothing happened. So now I’m confused.
Is there anyway to use the 2 wires that go from the transformer to the columns, and if so, how can I find info on how to wire them to the video doorbell? Am I correct in thinking that both posts on the transformer are positive?
As you can see, I’m pretty confused. I would like to have the video doorbells ring my existing mechanical chimes if possible. And I’d like to use the wiring that goes to the columns to power the video doorbells.
I’d appreciate any help and also, which Ring video doorbell should I buy that comes closest to my need. Thanks so much.