, and further tugging got it into 2nd, then eventually 3rd and 4th. ..........., and shifting was factory perfect with the old box.
I think I know what's happening. It is common to find the bores for the two forward shift rods to be badly worn out when the trans core is opened up. Especially if your gearbox was built from a core that lived it's previous lives in the land of perpetual summer. To correct this, the gear carrier is machined to accept bronze bushings. After the bushings are installed, they are honed to fit. The fit just has to be a light slip fit. You likely have a gear carrier like this, and the shift rods are just a little stiff. Once lube gets on them, and they get some usage, they will be fine. Your old gearbox shifted easy because it had more wear.
It also sounds like you are just tugging and pushing on the shaft that protrudes from the nose cone. Once installed, you have quite a large mechanical advantage when you hook up the shifter.
... the neutral part of the gate is very narrow, and 1st and 3rd are very close to it compared to 2nd and 4th, which are a normal distance back. ....
The distances you describe are not adjustable in the gearbox. The distance from neutral into any gear is exactly the same for all gears.
Different shifters will change the throw.