My brother has a 2001 4 cylinder 2.2 Rodeo Sport, 2wd, 110K. He got it 6 months ago. The oil warning light started to constantly flicker a week ago.
He's not big on maintenence/mechaical stuff but I'm pretty sure that he's topped off the oil when its been low. He hasn't changed the oil since he got it which means at least 5k but most likely more(who knows about the previous owner) on the same conventional oil.
Last night, he had a cousin change the oil. He drained a bit of oil to use a Gunk motor flush, ran the engine, drained the oil, then refilled with roughly 3.5 quarts of oil and 1 quart of Lucas oil stabilizer. Very black oil, I think there was even a "plop" sound of a small sludge-ball that we heard hitting the oil drain pan during draining. Started the engine and ran idling for about 15 minutes. Cooled down for about 20 minutes.

When he left he said that the engine was sluggish with a top speed of around 30mph before finally dying completely about a couple of miles away. It wouldn't start up and sounded as if it were flooded. It did turn over but wouldn't start.

Some other notes, increased ticking from the lifters before he got the light that never went away, used a work light powered by the auto's battery, the oil pressure indicator light never went off, he had the valve cover gasket replaced a 3? months ago because of oil in the spark plug tubes (not sure if that is still an issue).

I was thinking maybe the oil pump was clogged/malfunctioning and all that "Gunk" treatment on really nasty oil gummed it up worse. Maybe the battery drained too far and the alternator was weak (stretch, as flashers were bright).
Any opinions?

PS I have the V6 version of the exact same year/model

From what I can tell, it sounds like you may be right, may need an oil pump.

Engine dead due to lack of maintenance. The motor flush probably loosened a bunch of sludge and crud that clogged up the oil pickup tube. Probably just hastened the inevitable.