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One way to summarize and simplify what you described is that, when a circuit's input-output function (time domain) is an odd function f(x) = -f(-x) even harmonics vanish.

Now, what are the likely factors that disturbs that relationship in the actual circuit?

Variations in static characteristics like Vth, RD(on), gm, of the transistors. Variations in dynamic characteristics like what you mentioned, of the transistors.

Imperfectly symmetric transformer windings, board patterns, stray couplings/capacitances, etc.

When you have a class AB QRO push-pull amp, the bias circuits and possibly even NFB on the final stage can be tweaked to vanish even order harmonics. I think that's a costly tradeoff for most QRP amps, though.

Now the need for the RFC on the center tap, what it does and how important, you can see all sorts of theories and a classic "no one gets fired by inserting an RFC there" arguments go back for many decades. One most frequently invoked argument is to prevent RF flowing into the power rail.