Pluging a tire

Did you use a probe to verify the direction the nail came out and to prep the hole? Personally I will not plug a tire unless it is an emergency. Instead I use a combination plug patch that goes in from the interior and is rated to last the life of the tire. Plugs have been known to fail and you cannot inspect the interior of the tire and seal the inner liner.
As an example, this video shows what most tire shops will do.
https://www.youtube.com/watch?v=BKwBSzikVrY

The first step is to use the round file to rough up the rubber and grind down the sharp ends of the steel wires. That file will be your guide as to the angle. When you let go of it, it will sit at the angle you need to push the plug in.

There are times I haven't been able to push a plug in. Even twisting it didn't help. What has worked is to push the tool in as far as possible, then I roll the vehicle backward by hand. The handle of the tool will hit the ground, then pushing the vehicle backward some more is easier than pushing the tool in. I don't recommend running the vehicle back with the engine because you don't have enough control. The entire plug will end up inside the tire.

If you have the kind of plug that's already sticky, those work without glue, but coating it with rubber cement used for patches will make the plug easier to slide in.

Don't use a plug on or near the sidewalls. The flexing will work them loose. I like to use both a plug and a patch. I consider plugs temporary to get you back home. Or, to say that a different way, they seal really well, so I ignore them for years. The best approach is to install the plug, then dismount the tire, cut off all what's inside the tire, then patch the hole. Patches are a stronger repair, but plugs keep water and salt out of the hole where it would attack the steel belts.