Is the Weighted Maximum Coverage Problem with frequency weights NP-hard?
If you allow sets to repeat, then you can show that this is still NP-hard by making all the weights equal. The idea is to reduce from unweighted maximum coverage to your version of maximum coverage. Given an instance of unweighted maximum coverage, where element $e_i$ appears $w_i$ times, let $W = \max_i w_i$, and for each element $e_i$ of non-zero weight, add $W-w_i$ sets of the form $\{e_i\}$. In the new instance, each element appears the same number of times, so all have the same weight; and one checks that the added sets cannot improve the optimal solution.

If you insist that the sets are disjoint, you repeat the same construction, but to each set in the construction (old and new) you add a unique dummy element. This increases the value of each feasible solution by exactly $k$.