When we model supply and demand, I can understand why when we introduce the conception of taxation, we can model it as a shift to the left of the supply curve, this is because the marginal cost of production increases; so firms want to charge more at any given quantity to make up for the tax, so to speak.
I can also understand why it could be a shift to the left in demand, say I usually buy 3 apples for \$12 each, now there is a \$2 tax; that's going to reduce the amount I want to spend (not including tax) to \$10, it's going to reduce my willingness to pay.
The issue I have is when we just say "well it can be modelled by one or the other," because the result (that the price demanders pay is the same amount higher than the price suppliers pay, i.e. P_s + tax = P_d) is going to be true in either case. But why is it NOT simultaneous? I could understand perhaps if it was, say, a sales tax; or VAT, something which may have only one effect (although I don't know if even this is true, u/Hou_Civil_Econ I believe is saying that here: https://www.reddit.com/r/AskEconomics/comments/qploog/why_does_taxation_not_shift_the_demand_curve/) but if that is the case why is it not the case that demand shifts left AND supply shifts left?