In those German texts I found descriptions how the pretension in a sawplate is checked. They bend the saw and look for a lightgap between the plate and a straight edge. When the plate is properly tensioned it will be flat when you keep the saw straight like you would in normal use. It is flat lengthwise and crosswise. But when you set the plate in a slow bend along the length, the pre tensioned parts get the chance to release part of their tension by bending differently. The sawplate gets a curved cross section. The sawdoctors have measured this with the light gap, an old technique. One of those documents mentioned a lightgap of 0.4 mm for a properly tensioned framesaw blade as used in a sawmill.
So I decided to have a look at the several old saws I have. I don't claim to be a sawdoctor, so take this with the usual grain of salt.
I have two Disston D8's from the 1930's. I have an older Spear and Jackson (no idea how old). And I have two Tyzacs, which also look rather old to me with the name punched into the plate, instead of etched.
First I looked at the straightness crosswise with the plate held straight. They were all flat or almost flat.
Then I bend the plate, pushing the toe against a stop on my bench and bending at the handle end. So I had a free hand to hold a straight edge against the plate, crosswise again, in the middle.
On both the D8's I couldn't detect much change, the plates still looked pretty flat crosswise.
The Tyzacs both showed a more convex surface on the outside of the bend in the plate. This corresponds with the information in the German texts for a tensioned blade. The convexity is not much, but I really have no idea how much it should be for a handsaw.
The Spear and Jackson has the most obvious convexity. I think I can even see how the plate is tensioned through the middle and along the back edge. I tried to shoot some pictures from the S&J. Hopefully you can see what I mean.
A picture with the sawplate straight, you can see that it is flat crosswise.
And a picture with the bend saw plate. Here you can see the effect of tensioning.
In use the two D8's both have the annoying flapping habbit on the return stroke. The S&J and one of the Tyzacs are nice saws. The last Tyzac isn't in working order at the moment.
So, overall, I would carefully conclude that this handsaw tensioning business is more then just a myth.