First paragraph.
I agree with you, but what if your tool is the correct size and run out is acceptable. Both inside and out side would be small.
Second paragraph.
This will work if your backlash has already been taken out by first moving toward the indicator.
Try moving away from the indicator then slide the indicator into position then move away again then back.
This way the backlash will show itself.
Third paragraph.
A 1 inch travel dial indicator or even dial calipers can be used, but I think that backlash is probably the problem.
Best of luck, you have to admit this is fun, even with the minor problems.