Turning off then on with a CRT monitor is sometimes at least a partial "cure" to some problems, since the decay of the magnetic field at turnoff and the buildup of the field at turnon acts to demagnetize the tube. Some CRT monitors (and nearly all "tube" TV monitors) include a "demag cycle" deliberately built in to the power systems. It's unlikely to have any effect on a flat panel monitor, since flat panels aren't magnetic devices.
For a flat panel, a full turnoff/on cycle might help to clear some "stuck bits" in the memory in the monitor itself, although the kind of memory used would have to be known to predict whether it could help.
My reference to the broken switch was based only on "if something's broke, it's probably been in use for a while."
The concern about age is mainly related to the tendency, in devices with many switches (logic elements) for some of them to "wear out" with continued age - and the presumption that only the "turned on" time counts in the aging.
So far as I've found in documentation, flat panel monitors haven't been around long enough to generate much comment on whether leaving them on constantly is better or worse than frequent on/off cycling. Even for CRT monitors, the question is "controversial," with advocates for both sides of the argument.
The most common kinds of flat panel monitors depend on changing the polarization of light passed through "liquid crystal" bits for each pixel. The liquid crystal materials are generally an organic material that can be "uncrystallized" by temperature extremes, although I haven't found any documentation on the transition temperatures applicable to common flat panel designs, and would expect the critical temperature to be somewhat higher than the panels are likely to see in any normal use/storage. Each pixel has to be turned on/off by a separate "switch element," which appears, usually, to be a diode operated as a zener switch. Failure of one switch - or several in a localized area - could overload adjacent ones(?) so progressive failure may be a "built-in" feature of these devices, if the failure is due to switch element failure.
The "switching layer" in most flat panel monitors is, by nature, flexible (they're called liquid crystals for good reason). This makes most flat panels very susceptible to pressure on the front panel, which - if excessive - actually can damage the internal bits. Most larger (desktop) flat panel monitors now have much "harder" face layers than in the past, although laptop monitors continue, apparently, to tend to have rather flexible face layers. Part of the distortion seen if you press a finger against the face is due to flexure, and there likely is an additional distortion component due to the "capacitance" of your finger. Excessive twisting, poking, warping, or other "mechanical stressing" of a flat panel can be destructive. Careless "cleaning" can leave scratches, and use of aggressive/inappropriate "cleaners" can be very destructive to the face layer.
The only reasonably well documented mode of failure I've seen relates to the "backlight" that actually provides the light that's switched by the LX layer in the common kinds of monitors. In most early flat panel monitors, a fluorescent lamp was used. I haven't seen enough details to know whether other methods have come into use, but for some (not many?) of these earlier ones it was possible to replace the fluorescent lamp as a separate component. The deterioration seen due to lamp senescence would be a general overall dimming of the display, and would not be of the kind seen in the present case. In an extreme case, a fluorescent lamp might begin to "strobe" on and off, but monitors - so far as reported - almost never reach that stage.
Even very good flat panel monitors commonly come with a disclaimer that:
"Isolated pixels may fail to light. This monitor was tested prior to shipment to confirm that the number of pixels displayed meets our quality standards. The failure of a few pixels is normal so suck up and live with it." [paraphrased slightly]