Feeling 4th axis stupid

I have machined very few parts in a 4th axis on a vertical mill....even fewer with a tailstock holding one end. It seems that if the tailstock is not directly in line with the indexer I get the same results as if I were turning a taper in a manual lathe by offsetting the tailstock. I am machining a round part square in the indexer with a face mill addressing the top of the part. When I make the first cut it is perfectly level to the table. When I index 180 degrees that surface now has .002" of runout.......in my head (here we go) it is because I machined parellel to my table.....but.....when I indexed the part, I did so around an axis that is not parellel to my table????? Have I overly confused the issue???? In a nutshell......I am thinking....the axis between the tailstock/center needs to be perfectly aligned with the indexer/collet. What then of the center I am drilling in the part?......Will drilling my center off-center affect the end result? I am thinking it will not affect it.......I am gonna find out soon enough....just wondering what insight the people who do this on a regular basis may have.

You're correct. Aligning the rotary and tailstock is identical to aligning the headstock and tailstock on a lathe.

drilling the center off of center may affect the part once out of the indexer, like a long skinny shaft that bent from the center and once released went back to its relaxed state which may affect what you cut.

you would also need to index to 180 and check your taper in the z direction

Put in a really long precision round stock in the 4th, one with ends center drilled. Then indicate it close to the 4th and index further out. Make sure runout is low throughout. Then run the indicator on the top surface to make sure the rod is parallel to the table. Then install tail stock in test rod. Indicate again and adjust. Now your setup is accurate to mount your part.