True - that's one difference, at least as things have "evolved". Bearing in mind that double-run whiskey isn't a rule, so to speak. Back in the old days a lot of the stills had thumpers which negated the need to do separate stripping/spirit runs (because the traditional moonshiners just didn't have time to run stuff twice), while today the use of column stills does the same thing (all of Barton's products, for example, are single-run from a 26 plate column).
I've actually played around with it both ways, along with a 3rd way that includes pulling a high, narrow heart cut of a single run that's been soured with previous low wines, precisely like the OP suggests, then treating the rest of the run as a strip.
In fact, as I'm sitting here writing, it occurs to me that you could do both, in virtually any combination. Pull a little from what's left of a strip, pull some more from your spirit run, mix 'em up and let her fly... You could spend endless hours... hell, you could spend a lifetime... tweaking and refining until you come up with something that's so solid it makes you weep when you drink it.
I mean, isn't that what the passion of our thing, here, is all about?
A lot of this flexibility, IMO, is amplified and accentuated by still design. The uniqueness of the your distillate and the uniqueness of your copper are closely related. For my part - and I know this doesn't work for everybody - I've made a very fundamental decision to avoid factory stills. I found a coppersmith, we've spent many, many hours researching, prototyping, playing around with different things (and just generally having a blast) and he's building our equipment. One of a kind. Or two of a kind, as the case may be.