For those as likes point-buy or array for stats, do you also give everyone equal funds to buy starting equipment, or is that level of funds randomized somehow (a la 1e), or do you just assign starting gear?
In general, yes, I do prefer equal starting funds. I may make allowance for something else, if a specific option is built around the idea that it works so. E.g., in 4e, Background+Theme filled a space pretty similar to 5e's Backgrounds, just with a little more mechanical impact (4e BGs were pretty thin, mechanically, in most cases, but covered more of what 5e puts into BIFT stuff, while Themes were thicker, but covered more of your "this was my pre-adventuring life" stuff.)
So I could see a 4e-style Theme that, instead of giving any kind of power, let the player pick one level 1 uncommon magic item. Perhaps something like "Legacy Inheritor: You come from blood that has already proven itself heroic...which can be a mixed blessing. On the bright side, your heroic ancestor or relative has given or bequeathed you something they found useful early in their adventuring career. However, you'll also carry the weight of your family name--people will have preconceived notions of who you are and what you behave like, and that might not work in your favor." Later features might include stuff to represent your relative's reputation, e.g. you might get a beneficial response from folks, or you might be called upon to immediately help with something (potentially distracting you from your goals...or damaging your heroic reputation if you refuse), or someone might have a vendetta against you because of your relative's actions.
But yeah, if a player isn't spending a feat or character option on it, I prefer everyone starts with the same budget as well. Maybe have like a final thing at the very end (meaning, money that can't be used for purchasing character equipment) that injects a bit of randomness for your effectively "pocket money" to start out? E.g. if you're of the Noble background, then you might get some baked-in silk clothes and other fancy/expensive stuff, and roll (say) 3d10, while a mere Sailor might only roll 1d10?