A trivial reason why not, is because we use a three-phase AC system.
If each conductor carries a 3rd harmonic, and each current is 120° phase shifted from its companion, then regardless of the relative phase of fundamental and harmonic, the phase of the 3rd harmonic is (120°) × 3 = 360° = 0°. That is, if we insist that each wire has the same waveform (a symmetry condition), then all harmonic currents will necessarily be in phase!
So you're just shifting the harmonics to the common mode (between all lines collectively, and neutral or earth). Which, is a way to transmit power -- but it's basically a single-phase system again, and, it introduces complications when we actually want to wire this to a transformer, and add a ground point for safety and lightning protection: now we have either CM voltage on the neutral, or CM current through it.
If we propose a single-phase harmonic system instead, we still have the problem that fundamental and harmonic(s) must be treated separately, because they will undergo different phase shifts and impedances along the line, necessitating many more line reactors to maintain waveform (and now we need to maintain not just amplitudes, but the phase relation between fundamental and harmonic(s), to keep the peak voltage consistent as well). And we lose the handy feature that 3-phase power is always available (it never drops to zero, power is available continuously), and has a rotating direction (making motors much easier to use, and more efficient).
And the whole thing is even further complicated by loads that either abuse the harmonic content (nonlinear loads like rectifiers), or literally don't know what to do with it (the harmonic energy just causes useless heating in motors, and maybe torque pulsation as well).
Transformers also dissipate more power at higher frequencies; there is some pressure already to reduce harmonic content for this reason. Mind, it's a small effect -- fractional percent efficiency -- but we're talking massive grid-scale effect here, and that means saving billions of dollars over the installed lifetime of the grid and components (a transformer might last half a century or more in continuous operation!).
We could also propose DC for distribution (not just HVDC links), but besides being incompatible with literally everything it touches (how do you make such a change without starting over completely from scratch?), this brings further complications of safety, from the static fields that birds landing on wires would experience, to the incredible challenge of breaking high voltage DC arcs (for purposes of basic line switching, on up to safety in case of line breakdown). Not to mention annoyances like electrolytic corrosion (when a conductive path through water or salt film is possible). AC has the advantage that, arc power dips to zero twice per cycle, making it much more likely to self-extinguish, and much easier for a fuse or switch to clear the arc. It's also electrolytically balanced: apparently underground substations (vaults) can become flooded as a normal operating experience, and continue to operate, just with a lot more power dissipation due to the water only moderately shorting things out.
Still, with all these issues, we do have the technology to adopt a DC distribution system; it's just that it's immensely more expensive to do so. Switching will vary between specialized mechanical types, and semiconductors (high voltage SiC MOSFETs and IGBTs are available); arc fault detectors could be deployed; transformation can be done with semiconductors and power converters; fault currents and surge voltages can be handled by just using a hell of a lot of semiconductors (cascode stacks in parallel); and better insulation can be employed on wires and connectors, and sacrificial or noble electrodes can be placed near connections where corrosion may be an issue (like the ringed insulators supporting wires on poles). These are all known, understood and available materials and techniques -- it'll just vastly increase the cost of the system, for nearly no improvement in efficiency, and a decrease in usability.
In summary: whether we go up or down in frequency, or use a mixture thereof, there are many more issues that pop up, ranging from theoretical necessity to economic convenience. The 50/60Hz range seems to be good enough; there would be merit in considering a relatively small change, perhaps 40Hz to reduce transformer losses, or 80 or 100Hz to reduce transformer size (give or take the loss ratios of newer materials available for transformer construction), or perhaps certain special ratios in that range that would be mechanically convenient (maybe 83.3.. Hz instead of 80).
In any case, making any kind of sweeping change at all, requires updating the whole system, so these are largely historical-hypothetical questions: what if the system had evolved towards a different base frequency, back when change was still feasible?