I'm pretty sure that if you could somehow have brought some of our great west coast Douglas Fir (a staple in my industry) to a medieval lord, you could have convinced him to ditch the local oak in favor of that wood for his gate. So, in a way, yes, although obviously given real world constraints the best choice by far is to stick with what grows in your area's forests.
The Black Forest in Germany is mostly pines and firs, including Douglas Fir, yet German medieval furniture is also mostly oak as are beams in castle construction. Clearly, it was a concious choice, and not just "oak is handy."