Top 3 strings of my Tele are "dead"

So your Tele has the Jazzmaster-type bridge (that is, the bridge that rocks back and forth on the two posts)? If so, it could be that the bridge itself is too low, so your top strings are not pressing down on it hard enough as they pass over, causing them to not ring out properly. I have a Tele with a similar setup (albeit without the Mastery bridge), and I shimmed my neck a bit to be able to raise the bridge and increase that angle while maintaining my action how I like it.

 

Straighten out the neck? Had a few issues from fret 12 onwards on my tele that I'd only previously set the relief on (must have settled in a bit after a few weeks).

Went from feeling dead with no sustain to perfect in a few small twists.

 

neither the mastery for offsets nor the mastery for teles is designed to rock with tremolo use.

 

Start with the nut, bad slots

 

First, a string tree will do nothing but help, as it increases downward pressure on the nut.

Second, rule out open strings. Is it only open, or is it noticeable with fretted notes as well?

If the former, it's likely the nut slots.

If it's noticeable both ways, make sure your action is high enough. Before it's low enough to buzz on uneven frets, there will be a deadening of the ringing note. The string catches a whisper of the high fret during it's elliptical travel. Not enough to hear, exactly, but it dampens the response. If this is the case, a fret level is in order. If you want to rule it out, raise the action a LOT and test.

While these are the things I'd look at with a normal bridge, those offset bridges, and bigsbys are notorious for not having enough downward pressure over the saddle. Again, you can test this by pushing down on the tail of the string, after the saddle, while you pick a string.

 

Yes, I just learned this for myself while researching a Mastery bridge for my Jazzmaster. Regardless, if it's that general sort of bridge then bridge height (and the break angle that creates) is a potentially relevant concern.