why doesnt europe make their own F-35?

The non-VTOL version of the F-35 looks fairly unimpressive to me considering the enormous amount of engine power. Speed, range, climb specs dont look anything better than what you get with an F-16. The original intention of this jet was to have a low cost fighter, but now it costs almost as much as an F-22. So Im thinking the only selling point of this jet is its low RCS. Russia/India have already reacted with their T-50 PAK jet, which looks like a much better dogfighter. And China has their J-20 which looks like it can hold its own against an F-35. So why doesnt Dassault, BAe, EADS come up with their own low RCS fighter which Im sure would cost a lot less to buy and operate and equal the flight capabilities of the F-35.