Twang, is essentially a technique, which the tele is quite good at bringing out. Basically one over-stretches the string a little when plucking it and then on release the elasticity of the string provides the twang. The solid string mounting on a tele lets the string reverberate, any guitar can do this, Gretsch do it well too.
If you don't want it to twang, then don't do that.
We can and do play jazz on the tele - no twang, same guitar.
I have an AmSer07, and it does not have that "vintage" bridge plate, it has the "modern" bridge with 6 sintered steel saddles (however it can twang almost as well as my Baja06). Hard to tell from that pic but those saddles look as if they are pulled right back. Is that the original bridge plate? Is it intonated properly?
Changing the pickups or the bridge will not take the twang out of a tele, but you can with your playing technique. Acoustic players often thump an electric way too hard, ease up on that a bit, let the amp do the work.
There's two kinds of people, those that hear the music and those that don't.