If your radiator is 100% taxed out due to flow problems, size problems, engine problems, whatever... and you add heat from the oil system, then, yes, it will run hot. Without reserve cooling capacity, there's no room to take on additional cooling load. In the first example the underhood temps are so great it's warping plastic shrouds. That sounds like a flow problem to me.
The main problem with cooling our cars is trying to do it with a 30 years old radiator with horizontal flues... 2/3'd of which are physically clogged. And the one's that aren't clogged are physically coated with an insulating layer of crud that keeps the heat in.
Keep in mind, internal trans/oil 'coolers' aren't actual coolers at all. They're more like stabilizers. Water comes up to operating temp faster than oil and transfluid. And the water is the only one that has a thermostat to speed up the warming process. Internal coolers help put heat INTO the oil and trans until they get up to temp.
If the output of your radiator is 160* it cannon possible cool oil down unless it's well above 160 itself. And if the oil is so ungodly hot it's overheating your water, you need an external oil cooler to put that heat into the airstream instead of nailing the cold side of the radiator.