The owner of the marine elecronics shop where we have purchased all the electronics for both our US boats told me that the only way one could be in danger from the radiation from a typical recreational boat radar like a Furuno would be to hold one's head against the radome of an operating radome for a week.
Maybe you're onto something here. "Oh no honey, it's fine, just lay back down and keep sunbathing... Just keeeeeeep sunbathing..."
I think you are mostly right. Since radar frequencies are well below the "ionizing" frequency the major effect is heating. (ionizing radiation will destroy cells, i.e. ultraviolet) In that case the average power is what is important. However, some argue that the brain and nervous system as well as the eyes operate on electrical impulses. The higher peak power can possibly induce voltages that would interfere with the normal functions of these systems. Even after 70 years of radar use this is still an open issue. I personally would not stand in front of a pulse radar but would be much less concerned with a CW/FM radar which has very low peak power.