I noticed that most push botting charts use a fixed calling range for the other players when deciding what to push, here is an example

Hero has 5BB or an M of 2 with ante and is in SB, folds around, 2.5BB in the pot. Push botting chart assume that BB will call with something like A2+, 22+, any broadways, some fairly small percentage of his range, so in consequence Hero should shove ATC.
But not lets look at this from BB perspective, he has the same push botting chart in front of him and knows SB pushes ATC, he has to call for 4BB for an 11BB pot, so he only needs 4/11=36% pot odds so he will call with ATC.
In summary SB assumption on the calling range of BB is way off so SB is not playing optimally.

The mathematically proper way to set this up would be as follows. SB shoves some proportion p of his range (where 0<p<1 and p=0.3 means push top 30% of range), BB calls with some proportion q of his range.
This gives SB a payout of p(1-q)2.5 if BB folds and f(p,q)11 where f(p,q) is the function that determines SB chances to win an allin if he has top p % of his range and BB has top q % of his range.
Now SB tries to find the best p to maximize this payout function and BB tries to find the best q to minimize it. Game theory says there should be a unique Nash equilibrium with optimal strategies for both players which can be computed explicitly.

In order to make this actually useful one would need a closed expression of a good approximation of the function f(p,q).
One can use Poker stove to find any given values of this function, say f(0.5, 0,2)= 0.417, if SB pushes 50% of his range and BB calls down with 20% of his range SB has 41.7% chance to win.

Is this approach used in some poker software?
If somebody can provide me with the function f(p,q) I can do the computations for shoving and calling ranges depending on stacksize and pot size
note that the same idea applies in situations with more than 2 players but the math gets even more messy.

looks like I got a number of reads but no comments or feed back so I will try to summarize the argument in a less mathy way

Suppose you are in the SB, you are short stacked and it folds around to you. Great, no we consult a push botting chart to figure out what range we can profitably push here.
The push bot charts that I have seen all work as follows: assume that fixed calling range for BB (possible several options, tight, normal, loose) and given this calling range there is a fairly straight forward computation to get your most profitable pushing range

The problem here is that the BB calling range should depend on your shoving range the same way your shoving range depends on his calling range. It looks like I just created a mess with circular reasoning but the game theory says that given your stack size there exists an optimal strategy for both SB and BB. That is there is an optimal pushing range for SB and an optimal calling range for BB such that neither can do any better by deviating from this strategy. Given some more poker data one can compute the optimal ranges explicitly.

Now my questions are:
Are there more advanced push botting charts that make use of this concept?
Are people interested in this model and the data that could come out of it?

There's been an optimal strategy done for HU which is essentially BvB, i.e. SAGE.

IMO If you're before bubble you shove ATC with an M of <4 because very few people call wide enough. Sure there's a GTO strategy, but it's not the most profitable at the small stakes.

thanks, SAGE does more or less exactly what I was thinking off

disclaimer - I am not an expert. But my thoughts are,

Ive seen charts for the HU nash equilibrium. Problem with the charts is that they assume villain calls optimally which is obv pretty far from the truth. Ive never understood how you cant be loosing value by open shoving top of your range rather than just value raising. Or maybe I just misunderstood how to apply the charts.

Im guessing you could work out a good approximation for the nash equilibrium dynamically (in software) if you have a good model for the calling/pushing ranges of villains based on stats, history, game flow,etc. Youd probably make a ton of money if you had some software that could do it in real time, but its explicitly banned by the t&c of the poker sites.

Think SAGE is a way of quickly getting an approximation of the nash equilibrium?

way too much use of charts, software and misc other crutches

Course maybe I am not using enough toys

I think game theory is pretty useless for pushbotting unless you are playing an expert SNG player HU. You should push based on what you think their calling range is and make a rough approximation in your head. I guarantee the random MTT drooler is not thinking "it's GTO to call q8s here!"