Interesting analysis, a lot more to digest but some early questions 
- I am not convinced why the equilibrium between issuance and burn rate should happen. AFAICT it’s assumed that the equilibrium will be reached, but I don’t see why the current system should converge to that outcome.
- The values obtained ("at equilibrium in case x, the circulating supply of ETH is y") then follow from that equilibrium assumption. It seems that what the analysis proves are statements that sound more like "Assuming burn rate b, deposit D and equilibrium of issuance and burn, the supply cannot be anything other than S", but written up as “The supply of ETH will tend to S given burn rate b, deposit size D and eventual equilibrium of issuance and burn”. What prompts this remark is that I am surprised none of the results somehow depend on the initial supply.
Do you think some of the relations between variables that you model could be adapted to a more dynamic analysis then? E.g., given initial supply S_0, at step n there is an amount burnt b_n S_n and an issuance c F \sqrt{D_n}, so S_{n+1} = (1-b_n) S_n + c F \sqrt{D_n}. Then add modelling assumptions:
- The burn rate b_n could be negatively related to the deposit size D_n (e.g., your b', or related)
- Validators look for net yield (their y when they receive yD_n amount of rewards per year minus the supply increase \frac{S_{n+1} - S_n}{S_n}) greater than some \overline{y}.
Plugging such kind of assumptions (examples before are neither well thought out nor exhaustive) into the dynamic equation, it’s not clear that the issuance = burn equilibrium would be reached, though to me this is what would justify calling the “burn = issuance” statement an equilibrium beyond just an assumed equality. Curious to hear your thoughts!