Great to hear. That's how that stuff happens. You're getting the tool, so naturally the sight moves just before you get the new tool.
Now, new rear sight installation. The original rear sight was super hard to remove. Rust/corrosion? Rear sight too large? Slot in the slide too small? I don't know.
I do know that installation of a replacement rear sight can also be an issue if either the sight or the slot in the slide isn't sized right, one for the other.
If the new near sight won't fit into place in the slide without fears that you're going to break something trying to get it in you can "adjust" the rear sight to better fit the slot by removing some metal from the bottom of the rear sight. I use a stone. I've got a couple knife sharpening stones that only get used for gun parts. Metal removal for fitting and metal removal to remove rough spots to make the action smoother. Some oil on the stone, the rear sight bottom down on the stone and move the sight back and forth on the stone to remove some metal. I'm always worried about removing too much metal so I only remove a little and then do a test fit. If I think it won't move into place (and stay - it does need to be tight, but not so tight you need a big hammer and punch to install it - that sight tool should move it into place so you can tighten down the set screw) then remove it and take a little more metal off the bottom of the sight and then test fit it again. Repeat till you get it right. You can take metal off but can't put it back on, so go slow and repeat the test fitting attempts till you get what you want.
And, when you go to the range, take that little allen wrench for the set screw and the rear sight tool so you can move that rear sight if you need to during the sighting in process.