Perhaps I'm misunderstanding but you can use the variables when naming clips on the timeline as well, not just when rendering. It's not perfect, as far as I know you can't control the number of leading zeros for example. But my point is you can rename them all in one go, using the Inspector.
The
%{Timeline Index} variable is of course for the whole timeline, so if you need an individual index per track you have to disable the other tracks and just rename all the clips on a single track at a time.
Note that when changing the name of a clip (or clips) in the Inspector, you won't see the name change on the timeline immediately, you have to click away from the text box to update the timeline. The variables will resolve to actual baked-in values as if you had typed the values in, they don't stay as variables so if you move a clip the index in the name won't change.

To use these sequentially renamed timeline clip names when rendering individual files, use the
%{Clip Name} variable.
I suppose it says something about the discoverability of features in Resolve that several people in just this thread are asking for something that already exists (even if it could easily be improved). But like I said, maybe I've misunderstood something and this isn't what you're all looking for after all.
Adding a
%{Track Clip Index} variable and the ability to control leading zeros for all number variables seems like a reasonable feature request. Like
%{Track Clip Index[4]} or something.