Saytzev and Hofmann elimination in E1
In the E2 reaction, the size of the base can affect the selectivity of the deprotonation, because the steric interaction of a large base with a hindered proton will go over a higher transition state than the interaction of a large base with a less-hindered proton. This is kinetic control.

I'm not aware of any practical case where the selectivity of an E1 reaction can be controlled by the size of the base. The reason is that the deprotonation of the carbocation intermediate will have a transition state that is very close in energy to the carbocation itself. I agree with you that the size of the base should have some influence on the transition state energy, but in practice, it seems that it has less influence than the stability of the resulting alkene.

Also note that often E1 is run under equilibrating conditions, where even if a less-substituted (less-stable) alkene is formed, it can be protonated to reform the carbocation, and the material will funnel to the most stable product. This is thermodynamic control.