From what I have read, E1 largely produces the more substituted alkene because of the fact that more substituted alkenes are more stable as a result of hyperconjugation. However, in E2 the situation is different for a number or reasons. One of these reasons is steric effects of potentially large substituents on the $\beta$- carbon which would promote formation of the less substituted alkene. This effect does not seem to be mentioned for E1 but surely it must still be valid, right? Do large subtituents or bulky bases promote the formation of less substituted alkenes in E1? If not why not?
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.