When I worked at Marlin in the Repair Dept , early 70's, I was shown by 2 employees there who had been there when the LCS was still in production.

They placed the bbls on the action and clamped the bbl'd action upside down in the vise,,action overhanging the vise.
The top lever was already assembled into the action and attached to the yoke.

The trigger plate w/triggers was placed onto the bottom of the action and positioned so that the Safety was correctly engaged.
The front and rear trigger plate screws were inserted and tightened up, BUT just enough to hold the plate in the correct position and give it a bit of 'bowed' pressure
over the still disengaged top lever spindle.
That top lever spindle will be just off center from it's hole in the trigger lever plate due to it's engagement with it's heavy V spring in the frame.

To get the Spindle over, a straight shanked small dia punch is inserted into the screw hole in the bottom of the spindle.
Sometimes you have to first use a tapered shank to move it over enough against the spring tension so you can get the straight shank punch inserted.
The straight shank punch is used so it affords the greatest strength in the small dia screw hole (4-40) and won't tilt on you in the next step.

With the trigger plate positioned correctly, the punch inserted and slight pressure now pushing downward at the point where the spindle will enter the trigger plate,
place a small cloth across the bottom of the action to avoid scratches (they used to simply use their work apron).

A screwdriver with a wide flat blade and a V filed into it was placed against the punch.
The 'handle' of the screwdriver was placed against your stomach.
Some had an extra large handle on the driver more like a small bitbrace handle to afford better control,,and it was more comfortable to use for us smaller guys.

Now both hands are free.

Right handed,,,your right hand thumb presses on the trigger plate to the right of the punch while your fingers reach around the bottom and hold the spindle from falling back out from position once engaged in the trigger plate
Your left hand thumb presses lightly down on the screwdriver blade to help guide it as you push it and the spindle into position. Your left hand fingers are gripping around the action and bbls .
You have everything securely under control.

You use your body/stomach strength to push the screwdriver/punch forward that small amt to line up the spindle with the hole in the plate.
When it's in allaignment, the spindle will 'snap' into place from the slight tension of the bowed trigger plate above and the constant finger pressure on the top lever from below from your right hand fingers.
You can watch the whole thing very clearly as you are pushing the spindle via the punch & driver. There's no nervous moments, no need to rush or panic.

Once the spindle snaps in place hold it there in place from below and run the screw into position..
If you don't support it, you can often push it right back out from engagement from that V spring again.

The Nick Mackinson/Hunter Arms tool & technique is about the same from the looks of the pic.
It just doesn't use the action screws for tension nor your stomach to push the pin into place.

There are often a few different ways to do these things and the factory assemblers and repair people came up with them after a lot of experience working with the guns.
Sometimes factories would have special tools or methods advised in manuals but then other easier or expedient ways evolved as more familiarity was gained.