Hopefully we did not do any damage tightening down.
The input shaft rides in there with no bearing support, nothing is unusual there.
The centering of the disc should not be any significant issue; you can use the alignment tool that the clutch kit came with, but even at that it is not necessary. You need to get the disc close but it isn't a precision operation you would expect. We aren't talking about thousandths of inches tolerance here, you have a good 1-2mm of play.
When installing these clutches at Toyota, our discs don't even come with alignment tools. While technically somewhere in the dealership there is a Toyota alignment tool that is designed to fit proper with no bearing, it is never used. The reason your alignment tool has that provision is because it is simply based on spline count and diameter, not the actual vehicle application.... that tool is used on many vehicles they sell
different clutches for.
When I do these I will usually just use a long screw driver to hold the disc in place while I set the pressure plate on and thread the bolts. I do not
tighten the bolts down all the way, just snug them in a criss cross pattern a few times so that there is drag on the disc. From there I take a visual of the disc location with two considerations:
How centered it is in the opening of the pressure plate (at the spring fingers)
How evenly placed it is when looked at the exposed areas of the pressure plate around the perimeter
With it visually in position, you are always fine. Again you have a good 1-2mm of play here. Once it is visually centered, tighten down the pressure plate bolts and put the transmission back on. I can honestly say doing tC/Corolla/Matrix clutches I have always done this (specifically because the OE tool has been missing since ~2004) and I have not had one instance where I had to try again. It always works if you just do a visual on it.
If the car had a pilot bearing making the input shaft placement be a precise requirement, this would not be the case as it would have to be absolutely perfect.... but there isn't, and so it doesn't.
Let me know if I was unclear on anything.