Is this a proof that SET COVER is not an NP-hard problem?
Weighting of elements from set $P$ of SET COVER problem can be obtained by solving a system of linear inequalities for a given $\varepsilon$ that involves $\{0,1\}$-matrix $I=[x_i\in P_j]:$ $Iw\geq\varepsilon\sum\limits_{i=1}^mw_i.$ If it is not feasible then $\varepsilon$-DENSE SET COVER is void. Found weighting can be easily removed by repeating elements of $P$ with respect to the weighting but the resulting problem will be the same. Feasible SET COVER could be reduced to dense one for $\varepsilon=\frac{1}{m}$ but generally aforementioned reduction could lead to void dense problem for constant $\varepsilon.$

In the geometric case when VC-dimension is finite the dense problem with constant $\varepsilon$ could be solved in polynomial time due to result of Haussler and Welzl "epsilon-nets and simplex range queries" claiming existence of feasible solution to $\varepsilon$-dense problem of length $$\frac{8d}{\varepsilon}\log\frac{d}{\varepsilon}.$$
Having upper bound on the problem optimum we could solve te problem by brute-force search. Here $d$ denotes VC-dimension.