I understand the basic (very basic) theory of how a meter works. What I don't understand is why a meter can't have an electronic built in calibration that avoids the need for a calibration solution.
Does this exist?
No and the fact that it doesn't should tell you that it isn't possible. The reason for this is that the meter is just a bunch of electronics (high, make that very high, input impedance instrumentation amplifier, A/D converter, microprocessor and display). These are subject to gain drift, offset etc with aging but modern stuff is pretty good in that regard.
The electrode consists of a pair of electrochemical half cells. Together they produce a voltage, when immersed in a solution at pH of
E = Ei +s*N*T(pH - pHi)
where E is the voltage produced at temperature T and pH = pHi. N is a constant (N*T is about -58 mV/pH at room temperature). pHi is the isoelectric pH i.e. the pH at which the voltage produced does not depend on temperature and s is a constant near 1 called the slope. Clearly s is related to the required gain and Ei to related offset and if the meter itself undergoes gain change or drift, those parameters will change but it is with the fairly rapid change of those parameters in the electrode that we are concerned. To calibrate a meter we must stimulate it with something at known pH (much in the way we calibrate a balance with something with known mass) at known temperature. The buffers are solutions that have known pH as a function of temperature. Assuming (as we do) that pHi = 7 taking measurements give us 2 values for E, one in each buffer, and from those we get a system of 2 linear equations in 2 unknowns which we can solve for s and Ei. With those available we can stick any observed pH at any temperature into the equation and solve for pH.
This is really the same scheme that is used to calibrate any instrument. With a thermometer we must stimulate it with a pair of known temperatures (triple point and boiling points of water), with a balance, with a pair of known masses (0 and another mass close to the upper end of the balance range), for a dissolved oxygen sensor with a pair of conditions at which the DO level is known, for a voltmeter with a pair of know voltages and so on.
