The oil pump is driven by a shaft that goes into the bottom of the distributor. I have seen quite a few of these shafts twist in half. I would pull the distributor and verify this first.
1964 Mercury Comet Caliente / 260 Well, I considered all the suggestions and went with Mobil 1 oil and Wix filter. The oil pump priming shaft was the actual problem as the pic shows. Just twisted in half. I went ahead and replaced the oil pump since you have to drop the oil pan to get to the shaft. To do that I had to chop off some exhaust pipe, which wraps around the pan. I'm gonna put duels on it anyway (son's car). I also had to jack up the engine from the mounts to clear the pan. Cleaned it up, replaced the parts, put everything back and it runs like a dream. I'll put up some video soon as I get the timing perfect. ha ha.. Anyway, almost there. I'll keep posting updates.