When I worked at a public aquarium and did the water quality (31 systems, every day with far more involved tests than our home kits - which puts things into perspective!), I regularly tested ammonia...nitrite not so much. But ammonia in an established system was a sign of a problem and we DID see ammonia in many "cycled" systems that had been up well over a year.
To think that no problem will occur simply because the tank is established can be a mistake, and with such an easy test, it would be unfortunate to miss it. We see ammonia in overfed, overstocked tanks that were exceeding the biological capacity of a tank. These were tanks that were properly cycled but the bioload or feeding exceeded the biological capacity resulting in ammonia. Also if treatment for a disease had impacted the biological filter. Depending on what filtration was available in the tank, it, along with pH, could also indicate a failure of water flow through the system which you might not otherwise be aware of.
I never assume that a hobbyist keeps a system exactly as I do...that their filtration is checked and proper for their needs, that their stocking and feeding is fine. I would suggest to anyone, especially with less than a year in the hobby, that they regularly test ammonia - even if their tank is cycled. It does not need to be tested DAILY after cycled. But I would say weekly at least if you are stocking the tank.
If you have a simple system, add no fish, are consistent with your feeding, then I suppose you can say you do not need to test it. I rarely test it in my tank - but I do have the kits to do it now and then, if I see corals not opening right, etc. It is one of the first questions I would ask and eliminate if there are problems in the tank.
I believe the following basic kits are needed for a reef system: ammonia, nitrate, pH, alkalinity and calcium. If you dose, for example iodine, you should have that. For all the information they can provide - and problems they can eliminate - relative to the overall cost of livestock and equipment in this hobby? Don't cut corners.
IMO test strips are inaccurate, as are many of the "master test kits" such as aq. pharm. brand. Digital readers are quite nice, but they don't eliminate some things- digital testers need to be calibrated for example.