I have had both the 500/P and the 600mm/5.6 and I went thru a similar thought process of use of gear/cost/quality. I was able to compare them together for a period of time, and while I very much liked the 600, the 500mm f/4P is sharper. The optical designs are essentially the same, except for the 500 having one more ED element while the 600 has one less total elements. I feel the extra ED element gives a modern crispness to the 500 that the 600 doesn't, although the 600 is perfectly sharp for anything I used it for, landscapes, surfing, video time-lapses. The 500 has a much more saturated and crispy nature, nothing obvious but it does work well for the surfing side of my super-telephoto needs with the additional potential for salt-spray to shoot thru.
Both are fine on Tele-converters, I think the inherent higher saturation/contrast of the 500 makes it better than the 600 with the TC-14b, for both images are very slightly degraded but entirely useable quality depending on your needs. The 500 with TC is easier to shoot than the 600 with TC, a full stop of viewing lost in an film SLR is more dire than with a Z6, in my case. The focus peaking of the Z series is an exceptional way to use super-tele's, and I primarily shoot the Z6 for any color or video needs and black and white film exclusively with mostly F4's or F5, sometimes F3P. Screen change is a must, I prefer the U screen in general for both but both the H and G series can be useful when used with both types of converters.
With the 2x TC-301 both work better than most people generally expect, at least if you do things well; extra sturdy tripod, sandbags, a low wind and high clarity atmosphere. I add in a small amount of rigging, a cheese plate with a couple of carbon fiber rods where I can support the rear of the lens. With the longer TC-301 the lens length really gets long for both, and for me it's easier to tamp down vibrations when I rig them this way. The 500 is much easier to use as a sports lens, one stop faster shutter speed available of course, and the size and weight distribution without the hood is better with the 500. Also a plus is the CPU contacts and the useful additional functions that entails.
For color the 600 had a softer feel to it, or in softer light the contrast was delightful, whereas in the same light the 500 would seem to 'punch up the shadows' if that makes sense. Subtle. Black and white film was a toss-up choice with strengths depending on how I was going to use the lens and the day. The 500 seemed slightly harsher in B&W, seemingly less 'kind' to overexposure than the 600.
Either one of these specific two lenses are exceptional optics and both can give you extraordinary images, I went the easy route and kept both (until now, see below). I got the 500 very cheap with good timing and the 600 was offered to me by a good buddy for even less, so it was easy to keep both at the time
I still have the 500/P on another island but I just lost the 600mm f/5.6 in my family house fire in the Lahaina Catastrophe. It was a fine lens. It most probably but not certain at this point has been melted, according to what I've seen from overviews of the rubble.