Is the Euclidean TSP weakly NP-hard?
You're confusing a polynomial time approximation scheme (PTAS) and a fully polynomial time approximation scheme (FPTAS).  Euclidean TSP has a PTAS, but it is not an FPTAS because the polynomial increases in degree as 1 / ε decreases.  Only an FPTAS is disallowed for strongly NP-hard problems, assuming P $\neq$ NP.