That's both an issue with the 'pro' construction and also the nature of those thinner necks. I don't know why Warmoth say their standard thin is like a normal Fender profile when it's almost identical to ESP's Thin U and nothing like Fender's most common C.
But yeah, get yourself a thicker neck and you'll get a fuller tone. It supports your hand better, too, so it should allow you to play better and more comfortably, so long as you can still get your hands all the way around it, of course. I'm a firm believer that one should always use the thickest neck their hands can fit around. Maximise your sustain and tonal breadth, minimise strain and effort.
I just got myself a neck from another parts company that comes in at 1.09" at the 1st fret and 1.31" at the 12th. I trust it shall be quite spectacular.