You get the oil flowing as it's supposed to, not through a thin bypass in the block that blocks the flow. I picked up about 10 PSI oil pressure after installing an aftermarket oil cooler --- previous owners had plugged the fittings on the block and removed everything of the oil cooler setup. If it gets really hot, oil pressure still goes down to 10PSI at idle, so I wish I had a larger and/or more efficient oil cooler. It's something I'll probably change sooner or later ...
I don't see the oil temperature, but I can see the relationship between coolant temperature and oil pressure. You definitely want a good oil cooler to remove heat from the oil and thereby from the engine as well --- and the stock setup using the side tank in the radiator is a very good thing (I wish I could use that).
I had to make use of the other side tank for the transmission oil because with a 11x11" B&M style transmission cooler with a fan on it, mounted on the front, there was no way to keep the return temperature at reasonable levels. The temperature would just go up, depending on load. Since the side tank is in use, I can easily keep it at 160F--170F even when it's 100F+ outside.
The side tank alone wouldn't do it, but it does a great deal of the cooling. So for the engine oil, you do want the side tank --- and eventually another cooler behind the grille.
Plugging the fittings and driving without an oil cooler is just a very bad idea. There's no way for the oil to get rid of any heat without it, and these engines can generate a lot of heat. If you still have the stock setup but only need new rubber lines because they are leaking, just get them and put them on. If I had needed only the rubber lines, I'd have done that ...