Australia

If your two middle paragraphs are listed in order of priority

They are not. They were merely in the order it came to mind, based on context. Since the thread is primarily about road design, it's natural that the existence of public transport should come first. That's also why I started with "bus lanes" first, and not light rail. Bus lanes most directly compete with parking lanes, while light rail tends to compete indirectly (bus lanes literally being located in what would otherwise be parking space, while light rail tends to run down the centre but possibly requiring removal of parking to enable car lanes to continue to exist).

That said, I reject the notion that it needs to be done in a particular order. That's a surefire way to ensure nothing ever gets done, because you might say you need density for improved public transportation, but someone else will say they won't get rid of their car until there is first good alternatives.

But also, while higher density is certainly necessary for cyclability, I don't even believe it really is that necessary for public transportation to be viable. Remote US towns were built on the backbone of train networks. Rural towns in Europe have better public transport than much larger cities in America. Yes, increased density makes public transportation even more efficient, but efficiency is not a necessity for it to be viable. Only the political will to have it be good is necessary.

So I support, very strongly, any effort to improve public transport or increase density, regardless of whether it is done before, after, or alongside the other.