You can't play a modeler, it's simply a signal generator following an algorithm.
It's not a signal generator, not at least in the manner of say a synthesizer. Take a look at the page I linked--those functions are way too simple to accurately model even a single tube, but they building blocks. So if you have, for example:
f(x) = ax^3 + bx^2 + cx + d
That is entirely a function of x, the input signal, and a few constants (a,b,c,d; think of them as "mix" controls). The squared term yields second-order distortion (preamp tube); the cubed terms odd-order (push-pull output into transformer), and the regular ol' x is the mix of the original signal (d would be intended to cancel offset).
As I said, too simple for a proper model, just a building block.
But it's not the same as a synthesizer at all, which is a time-based function: a MIDI signal triggers a tone generator:
f(x,t) = sin (x,t)
which has to feed an ADSR envelope. Much more complicated, and yet musicians are able to play synthesizers, aren't they?
This gets back to my other thread: nobody that is playing a tube amp that possesses all of the time variant and invariant nonlinearities is playing anything like the natural sound of their instrument. Why not plug your guitar into a very clean, very high input impedance FET DI into the board and PA? Doesn't that sound good to you? Does that react in all of the nonlinear ways you ascribe to a tube amp? Of course it doesn't. And actually there is nothing special about the FET; I could give you a 12AX7 DI that behaved in the same manner. But that's exactly the way I play bass (although I just use a transformer). And actually, I record guitar that way too, just listening to the natural sound of the instrument. Later on while mixing I'll mung it good with various plugs, but I don't monitor with them.
Thus I think we need to dispense with the idea that it's impossible to play through any type of modeler because they are synthetic. The modern guitar tube amplifier is synthetic too; it's an analog additive synthesizer triggered by an input signal. Look at something like the Mesa rectos and you see all kinds of features purposely intended to alter that synthesis. It's more akin to a theremin or a Hammond organ than an acoustic guitar.