Given these reports I would not risk a nice NOS 5Y3 in one of these amps - it's clear that the circuit demand is right on the edge of what they will take, and you may kill a perfectly good tube that will run fine in an amp which is more correctly designed to run with a 5Y3.
I would use a 5V4 or another of the higher voltage and current rated types, depending on how much sag you want - the 5R4 is actually the closest in forward voltage drop to the 5Y3, but is rated for double the current and voltage (actually more than double the voltage, for the GYB variant), and draws the same filament current. This should be bombproof in a 5Y3 application. You could also use a 5AR4 (GZ34) although you'll get a lot less sag. Of the other fairly common types I would avoid the GZ37 and 5U4, even though they have plenty of voltage and current capability, because they draw a lot more filament current.
For what it's worth I do like the 5V4 too though, I have a pair of NOS Sylvania ones in my Tremoverb - which is just OK on current (350mA total for both tubes, at the upper limit when it's really cranked), and they sound great and have been totally reliable.