ELI5: Why does adding Pitch when checking threads matter?

using pitch 'ensures' that the probe always hits on the same portion of the thread, be it the very peak or somewhere else, but it is the same relative place on the thread. No pitch and opposing hits will be (for example) on the peak and in the ditch. not good for size or location.

This depends on the probe diameter and the physical size of the threads.

--If you have a 3" diameter/ 4 threads per inch (0.25" pitch), and you're checking it with even a 5mm probe, you can hit the Min dia of the diameter and still fall a solid >0.100" into the thread when you hit in between the teeth, rendering your diameter's accuracy = garbage.

In contrast
--If you have the same 3" diameter bore/boss with some custom 28 threads per inch, (0.035" pitch), and you're using the same 5mm sphere, your sphere can only dip less than 0.010" in between the teeth... adding some bias, but if your minor diameter is permissible to be 0.080" it's basically noise.

If you want to mitigate diametrical and positional uncertainty, add the pitch.
--Regardless of where you hit on the threads, you will get an accurate position... as long as pitch entered, and pitch cut is similar.

If you want to one-up your thread checking effort, you can run a series of hits along a line, on one edge of the threads, that has set increments (say 0.005"), and a length of about 2 pitches... then sort for the max or min (boss vs bore) PR of that series of hits... you can locally origin your Z axis to that max/min hit and always take hits on the thread major/minor diameter (again boss vs bore).
--Take your circle with pitch from the z origin and you've got super accurate major/minor diameter and location... regardless of where the thread start is clocked.

if I have a pitch of 13, (~.077), and I check it using a cylinder measuring it at 3 locations, then those locations need to be relative to the pitch as well correct? So I would need a length of ~.233 and lets say I want to be a few threads below the surface so I should check it at a depth of ~.154?

All those factors matter correct?

if you do add a pitch make sure you select CCW not CW. Using find hole & 3 pt face feature really helps to find the true center, depth & pitch approach.

KatmandudaMeow noted!

Yes, any pattern of circles should be an integer of the pitch.
IE circle 1 should be at least 2 threads down to eliminate the lead thread.. so start at 0.077*2 = 0.144 in z
circle 2 could be about mid-thread (let's say your threads are 1.5" deep), so 13 threads per inch at 1.5" long is 19.5 threads. (round) half of that is 10 threads * 0.077" = 0.770 in z
circle 3 could be about the same as circle 1 (2 threads prudent of bottom) so 16 threads down? 16 * 0.077 = 1.232" in z