Why does a protic solvent favor E1 over E2 mechanism?

protic solvents all all have an OH group which can form hydrogen bonds (intermolecular forces) which will surround the nucleophile in E2 reactions and trap it. Thus, the reaction will not take place as fast. This is because in E2 reactions the rate depends on the concentration of the nucleophile as well as the substrate's conc.. In E1 reactions, the rate only depends on the substrate's concentration thus trapping the nucleophile doesn't affect the rate at which the reaction takes place and since protic solvents are more polar than aprotic solvents it would prefer the more polar solvent. This is why u can use a weak nucleophile/base in E1 reactions but need a strong base in E2 reactions. This is the same concept for SN2 and SN1 reactions as well.