I'm of a way of thinking, that with wildlife and birds in particular, reach is important. I have the Sigma 150-600C, that I use on a 5Div and also, on an M5. On the M5, the lens is 960mm equivalent at it's longest.
I say reach is important, I used to think it was everything but it's not. In all honesty, after the last year's experience of birding, I'm begining to realise that light is just as important. I really wish I had bigger apertures than f6.3 (on the Sigma @600mm) more often than not these days. Living in Ayrshire, light can be a struggle as it's prone to being dull and grey here.
My thoughts and feelings now are that reach is handy, very handy but it isn't everything, I now believe that aperture, in our climate at least, is as or maybe even more important. Don't get me wrong though, I love the Sigma, it is a superb lens for the money and sharp too, I cant see me parting with it any time soon.
If it were me now and money was no object, I'd be getting at least an f4 or even better, an f2.8 prime. My budget would dictate the focal length and I'd buy the longest I could afford but it would be the aperture I'd consider first, for what I do. Despite loving the Sigma, it would be Canon L, as I think the 5D's sensor deserves it more than 3rd party glass. I have some L glass but nothing over 400mm and f5.6, although I do have the 2x mkiii TC. These days, I find myself worrying about the aperture over focal length. For sure, you need reach ( aboslute minimum of 300mm and a TC when needed) but overall, I think it's a balance between aperture, focal length and budget.
I'm curently mulling over my options ready for next year, so many lenses make sense but at a minimum, it would be the 300 f2.8, with my TC, which would become 600 f5.6. Ideally though, the 400 2.8, 500f4 or the 600f4 but any of those would have to be early versions and used, as £13k for a lens is difficult for me to justify, let alone afford.
Also, getting closer to the subject physically, whilst challenging (but part of the fun), can be done and can buy you more than any amount of reach. They also don't have to be frame fillers.
I appreciate your budget though and that might limit your options as a
second hand prime alone could easily swallow £2k.