Is there an intuitive explanation for the tax incidence formula from elasticity?
Ah, never mind, got there.

Both supply and demand share the same change in quantity as a result of the tax, so

$$\Delta Q_S = \Delta Q_D$$

The amount by which the quantities changed is the derivative of that quantity wrt price, times the change in price that side experienced

$$\frac{\partial Q_S}{\partial P_S}\Delta P_S = \frac{\partial Q_D}{\partial P_D}\Delta P_D$$

Multiply both sides by the equilibrium P/Q

$$\frac{\partial Q_S}{\partial P_S}(\frac{P^*}{Q^*})\Delta P_S = \frac{\partial Q_D}{\partial P_D}(\frac{P^*}{Q^*})\Delta P_D$$

which of course is the formula for elasticity at the equilibrium

$$\varepsilon_S \Delta P_S = \varepsilon_D \Delta P_D$$
$$\frac{\varepsilon_S}{\varepsilon_D}\Delta P_S = \Delta P_D$$

So then the producer's share is 

$$Incidence_S=\frac{P^*-P_S}{T}=\frac{|\Delta P_S|}{|\Delta P_S|+|\Delta P_D|} = \frac{|\Delta P_S|}{|\Delta P_S|+|\frac{\varepsilon_S}{\varepsilon_D}\Delta P_S|}$$

the price change cancels out and you get producer's share as

$$Incidence_S=\frac{1}{1+|\frac{\varepsilon_S}{\varepsilon_D}|}=\frac{|\varepsilon_D|}{|\varepsilon_D|+\varepsilon_S}$$

So it all comes from the fact that this is the unique formula that produces the price-change ratios that generate identical quantity changes in both sides of the market. Probably not any intuition that can come down to a principles course level but makes a lot more sense than it did before!