I very much doubt that the whole plane 'jumping' or 'skipping' would give the effect Sellers experienced. The high-pitched squeak (and a pattern of little cuts on the workpiece about 1/16" or so apart) suggest a much higher frequency event than the normal human could induce just by pushing a rigidly set up plane over a piece of wood. I think the result of that would be marks much further apart. It can happen if a plane is set far too rank, or is significantly dull, or both, and usually is something people only experience in their very early days of woodworking, before they've become familiar with plane use.
The Sellers chatter is down to a thin iron fluttering. Here's how.
Consider a standard Bailey-type plane in normal factory spec - no after-market thick irons or cap-irons. When dissembled from the plane for sharpening, the iron is flat. When the standard bent-metal cap-iron is attached, it takes a very slight curve, the cap-iron contacting the blade over a fair area near the screw, and along a line just behind the cutting edge on the flat side of the blade. Between these two points, there's daylight between iron and cap-iron. Thus, when the iron and cap-iron assembly is installed in the plane, and the lever cap pressure is applied, the blade is in close contact with the top of the frog casting, held by the top of the lever cap. However, the contact point (line, really) at the bottom of the assembly is along the back of the bevel, or along the lower front edge of the frog if the latter is set forward of the sole aperture. Thus, the lever cap pressure, transmitted through the cap-iron, acts outboard of the contact between blade and sole or frog, thus tending to flex the blade even more. All this can readily be observed on any standard Bailey plane.
Thus, there's detectable daylight between frog and blade, and between blade and cap-iron, between the back of the blade's bevel and the top of the frog. Again - this can be observed, and checked with feeler gauges. The blade is unsupported between these points - not trapped and rigid.
Now - let's put the plane to work. As soon as the cutting edge enters wood, there's a force needed to push it forwards and take a cut, and thus, an equal and opposite force pushing the blade backwards. Because the blade is bedded at 45 degrees, pushing it back also pushes it slightly downwards, deepening the cut. The blade pivots about it's contact point at the sole (or bottom of frog) along the back of the bevel. It's got room to flex between the bevel and the top of the frog, since it's not firmly held, so it flexes, allowing the cutting edge to go a little deeper. For normal planning that's just about OK, but in very hard woods or on endgrain, the force is enough to flex the blade so far that enough energy builds up in it (as it does in a spring) that it eventually has enough to overcome the force applied to the cutting edge, and it springs up, out of cut. Since the plane is being held down to the wood, it immediately starts to cut again, and the cycle repeats. From observation, the frequency must be somewhere in the 50 to 100 cycles per second region, which is why we hear a squeak rather than a series of judders.
The effect, as we know, can be avoided by using a stiffer blade, or by stiffening the blade with a Stay-set type cap-iron. That transmits the lever cap force to the blade at three points - the same two as the bent metal cap-iron, but also, crucially, a third point at the joint of the cap-iron. That holds the blade tight to the frog, giving it much less scope to vibrate or flutter.
This can be demonstrated with a six-inch rule on the bench. Place the rule at right angles to the bench edge, with about 10mm-12mm (3/8"-1/2") overhanging the edge. Place a finger firmly on the inboard end - the clamp between top of frog and lever cap. The bench edge represents the back of the bevel, the rule end the cutting edge. Now apply some 'cutting force' to the rule end, and the 'cutting edge' deflects down. A point at about 2 1/2" also pops up, quite a bit. Now apply a finger to about the 2" mark, representing the joint of a Stay-set cap-iron, and then apply the same 'cutting force' as last time to the rule end. It's now far harder to deflect the rule at all - everything is much stiffer.
That's how 'chatter' or 'blade flutter' happens, I suggest - and that's how to avoid it, too. It conforms nicely with the effects many people have reported using different blade and cap-iron combinations.