Newbie needs help with starting!

YAMAHA TDR 125 2002 is low to start with the choke.All new parts.
So my son got his first bike and he loves it but starting it is a pain.Electric ignition.No kickstarter.

I wont go into the long list of woes we had with it but we have figured that the previous owner didnt really put much into service wise.
My son and I decided that we will keep it and get it back running great and put the money into it. He loves the bike so wants to keep it.

So the main problem is starting.
On the choke its very hesitant and really hesitates to get going.
Once going and warmed up its fine and will re-start every time. Its the cold start on choke is the issue.
I put a new/refurbed carb into it and it started fine but didnt run right*.Then started to leak fuel from the overflow pipe. I adjusted the float height and this cured that issue and then was fine but cold starting on the choke is becoming an issue again.
I have tried all positions of the adjustment screw (I believe the yamaha setting is 1.5 turns out) and none of them give a good strong start.
The screw is located on the rear of the carb so presumably it controls the air.

Any pointers?
TIA




*I discovered that the reed valve gasket was in bits and the reed valves were not in good shape either so were replaced and made a big difference. Everything has been replaced from the petcock valve up to the plug so is effectively a new bike from that end with the exception of the rubber boots coming in and out of the carb and they look in good condition.