Any reason not to use a switch to select output impedance?

I think this is what the OP wanted to clarify, you say "16 ohm tap and common (0Ω) on the secondary" and Tom say "From the 16 Ω tap to the ground tap on the secondary of the output transformer".
You wouldn't use that because that ground has nothing to do with the signal in any way that circuit. That is why the ground connection to the OT secondary is invalid. Its a fully balanced negative feedback and would not need the ground connection at the speaker. Amps with unbalanced negative feedback like guitar amps are the only ones that need a DC ground connection to the OT secondary's winding.
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I think this is what the OP wanted to clarify, you say "16 ohm tap and common (0Ω) on the secondary" and Tom say "From the 16 Ω tap to the ground tap on the secondary of the output transformer".
Correct.
I guess Tom doesn't realize that ground connection its just a static chassis ground and not part of the amplifier circuit.
I heard of people grounding the speaker secondary before, but its bad practice when its not part of a feedback circuit. Some on the older multi tapped radio transformers, you did that, but that is because the speaker windings are not symmetrical.
Guitar amps use an unbalanced negative feedback, so that is why they have theirs grounded. Other amps like VOX did it to their combos so it would pop a fuse quickly if the combo amp got rained on.
 
Stupid question, what do you mean by “100 ohm across the 16 ohm winding”?
The resistor is connected between the common lead of the OPT and its 16 ohm tap. It preloads the xfmr to reduce voltage transients if connections to the transformer are changed while the amp is powered up. The resistor does waste a small amount of power, but the transformer protection afforded by it is deemed worthwhile.
 
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The value is not terribly critical, as long as it's a couple or few hundred Ohms. In my last build I used 470 Ohms between the 16 Ohm tap and common. I could also have used 235 Ohms between the 8 Ohm tap and common, or 117.5 Ohms between the 4 Ohm tap and common, with the same effect. Too large a value and it doesn't solve the problem. Too small a value and you start reducing the load impedance more than necessary.

EDIT: clarification
 
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So in this case, that would be from the 16 ohm tap to the 4 ohm tap. What's the significance of the resistor value? I've seen suggestions from 100 to over 100k.
I would pick the lowest value of shunt resistor that does not cause a significant waste of power. It's best if the resistor is placed across the entire winding, rather than just a portion of it, so it would go from the common or 0 ohm tap to the 16-ohm tap in your example. I'd go with something between 100 and 200 ohms. If you know with certainty that the speaker will never be disconnected while the amplifier is in operation, then I'd omit the resistor. That way you waste no amplifier power.
 
Did you mean to reply to my question about resistance, not connection?
No. I meant that which connection you use changes the effective resistance seen by the transformer, but not enough to matter.

Let's say you connect 470 Ohms between the 16 Ohm tap and the 4 Ohm tap. That's the same as connecting 1880 Ohms between the 16 Ohm tap and common. (There are exactly the same number of transformer windings between the 4 Ohm tap and the 16 Ohm tap as there are between the common tap and the 4 Ohm tap.) Is that better or worse than connecting 470 Ohms between the 16 Ohm tap and common? I'm saying that it's not enough difference to matter.
 
So the resistor could be connected between any two taps and accomplish the same job?
Within reason. Don't purposely push the boundaries. But if your intent was 470 Ohms between common and 16 Ohm taps, but you only have 220 Ohm resistors, then connect them between common and 8 Ohm taps for approximately the same effect. Be thoughtful, but don't obsess over getting exact values.
 
On this amp, the 4 ohm tap is the common (apparently, I'm just repeating what I've been told), so wouldn't that be the second connection point?
4 ohm tap is a ground reference and the ground is not part of the balanced signal. If it was, then you couldn't use the 4 ohm tap.

DC ground is not a valid signal common in this amp.

the 0 and the 16 ohm tap is where I would load it and within the 3 to 5 rule standard for output loading that most of you guys don't know. That is why I picked my values for resistors from (75 ohm or a 62 ohm). Because too high of resistance isn't going to load it enough and arcing may happen in the output tubes.

Since output transformers secondary is made from two 4 ohm coils and a 8 ohm coil tied in series, If you just go across one coil set, you have to calculate the resistor for the 4 ohm or 8 ohm coil. 0-4ohm, 4 ohm to 8 ohm, and 8 ohm to 16 ohms is the three secondary coils.

If the ground was really part of the signal, the ground reference would have to be connected at the speaker output's 8 ohm tap because that is the center tap of that coil assembly.
 
What does “common” mean, generally? I was conflating it with “ground”
Common is the AC "ground" which is zero crossing of the signal. That is why the wiring consists of a non inverting signal and a inverting signal (twisted pair)

In a fully balanced circuit, the DC voltage source(s) and the DC ground are not referenced with zero crossing of the signal. So the circuit gains about 40db of noise rejection due to this.

In an unbalanced circuit, the AC "ground" is referenced with the power supply, So all DC voltage sources and the DC ground is an AC signal common. The unbalanced circuit adds the power supply noise into the signal. This is your common build type because of it being half the parts and no hand matching is required.
 
the 0 and the 16 ohm tap is where I would load it and within the 3 to 5 rule standard for output loading that most of you guys don't know.
I'll admit that I've never heard of that rule. But I'll also add that this resistor is intended only to save the amp under unusual but temporary conditions -- in this case while switching between output taps; in others perhaps the speaker wire has somehow become disconnected. (Though I do realize that the first instinct, when audio is not coming out of the amp, is to turn it up!) Your recommendation of 62-75 Ohms is lower than any other recommendations that I've read (typically they say 200-1000 Ohms), but it's still high enough that it shouldn't affect the load impedance in any significant way.