First, let me apologise if I have caused some confusion. The center wire I am referring to is on the 6R7 side, not the 6V6 side. The 6V6 side does also have a center wire as shown in the diagram below and this is definitely a center tap as confirmed with a continuity test. However the 6R7 side also has a center wire, which has no insulation and, as mentioned, was connected to the chassis. This wire has no connectivity to either end of the 6R7 or the 6V6 winding.
Franz, thank you for posting the drawings. These are very helpful. I presume that the first one is from the original SX16 circuit diagram? This is interesting as the anode of the 6R7 is supplied directly with B+ via a resistor rather than through the transformer winding as well, as shown in the SX17 circuit:

The capacitor then blocks DC going to the transformer but passes the AC signal. I had wondered whether it was possible to design it that way and not have B+ going through the transformer winding. I do wonder whether that fact along with some fault, possibly around the 6R7 circuit (the voltages are a bit off around that and the silencer) caused an excessive current flow which caused the transformer failure. If that can be avoided then so much the better.
However I am also curious about your second 'Hammond' drawing and why you chose the 124D rather than the 124A? The reason I ask is because my initial inclination was to choose the model that closest matched the DC resistance of the existing transformer while maintaining the same ratio, which the 124D does. However, I was also advised to just use the cheapest option, which is the 124A. This is significantly cheaper than the 124B or the 124D.
Although supplying the 6K7 rather than the 6R7, R35 in the above is 10k, which matches your Hammond drawing, but I see the first (SX16?) drawing uses a higher resistance value of 27k? Does it matter so long as an adequate current is maintained to the 6R7, or whas there a specific reason for choosing 10k? Incidentally, in that second drawing the anode of the 6R7 does not appear to be receiving a DC feed? I assume that is just a drawing anomaly?