Toilet tank slow leak such that it runs for a few seconds periodically to refill
Hoofbeats = horses
Which is to say a common problem like this is still probably the flapper rather than some exotic unicorn of a problem.
Here is an experiment:

Get bricks or something comparably heavy and inert.  Put bricks on top of the flapper in the tank pressing it down.  That thing will be sealed. Beet juice or food coloring or not as you see fit.  If you don't get any more leaking then it is the flapper somehow.

Why might you still not get a good seal with a new flapper?  Maybe the bottom of the toilet tank around the hole is uneven - hard water deposits?  A good first step for stuff not working as it should is to clean it.  I will testify that one need not be a mechanical genius to clean stuff and have it work better.  I would start with scrubbing the bottom of the tank where the flapper sits when closed.  Maybe scrub with an old toothbrush or scouring pad.  See how it works then.
Without the bricks.  See how it works without the bricks.  Reserve those for later use.