Refrigerator cools once but then doesn’t start up again

It's probably just a twitchy power switch (spoiler: it was). Replace it or, if it can be cleaned, cleaning might fix it. For sake of completeness: Try to determine if the power switch is a simple switch or if it is connected to some kind of built-in circuit breaking device. If a circuit breaker somewhere in the refrigerator is being activated, and reset when the power switch is cycled, the breaker could be faulty else it is receiving out-of-specification power from somewhere "upstream" (e.g. house wiring or a faulty component in the appliance).

Otherwise, use a multimeter to check for electrical fault in the refrigerator: If there is not continuity across the power switch when the fridge has "gone off," repair or replace the switch. If there is continuity, test voltage and continuity "downstream" from the AC plug until you find a fault. An electrical diagram of the appliance may help.

Thanks to mjohns and jPhi1618 for comments that formed the bulk of this answer.

three main points come to mind (in this order):

1. Door seal(s): even though you may think the door is closed all the way, a weak magnet or bad seal will cause this problem EVERYTIME. Try the "dollar bill" trick: place a dollar bill in the area around the door seal and do a full series of tests the full perimeter of the door. The dollar bill should be held in place by the door seal when closed and should require effort to pull it out. If it is "easy" anywhere, then the seal is poor and moisture is getting in greatly accelerating frost buildup
2. Air fan: While you identify the fan is working, is it working properly? The motor itself can be marginal, running poorly, or not at all, check contacts and wiring, and also confirm the bearings are firm but the blades spin freely. Also (and this is critical), ensure the supply and return ducts are not blocked. Make sure they are fully defrosted, and clean. Any food, mold, plastic bags, etc are removed and do not store products in front of or blocking the ducts
3. Refrigerant cycling: if there is a flow issue (pinched lines etc) or too low refrigerant, the unit can be cycling on and off or never off, either way allowing the ice buildup, but this is usually only found in old or damaged units. Try setting the freezer to its warmest acceptable level, and make sure it is cycling off vs. running 24/7

As noted in the other answer, but amplifying it a bit further, make sure that if the unit has a defrost tray that it has water in it - it is possible the ductwork is bloxked by plastic debris or mold/food. Make sure all tubing is free and clear