It's first worth mentioning that the Sage Advice you cite is now obsolete; errata have been released which contain a correction so that long rest does now require sleep as a general rule:
A long rest is a period
of extended downtime, at least 8 hours
long, during which a character sleeps for
at least 6 hours and performs no more
than 2 hours of light activity, such as reading, talking, eating, or standing watch.
The Basic Rules (which still give the original version) and the PHB (which is updated in newer printings and on D&D Beyond — payment required) don't say exactly what happens if you continue to not sleep, other than not getting the benefit of a long rest, but
Xanathar's Guide adds rules for this: DC 10 constitution saving throw to avoid a level of exhaustion after 24 hours of being awake, and the DC goes up by 5 for each additional 24 hours.
For some reason, there's no level of exhaustion between "so tired you can't move" and "completely dead". It seems like at some point "you fall asleep" would be on the chart, but, nope. Death it is.
With that in mind, a Con 10 character without proficiency in that save has a 16.5% chance of being fine after 48 hours, but only a 0.82% chance of feeling normal after 72. (Of course, a character who made it through the first 48 has a 5% chance of one more night where everything seems fine.) After that, though, the DC is impossible, so it's just a matter of counting down to sudden, permanent sleep.
You'd think a level 20 barbarian with 20 Con would be much better at ignoring the pesky limitations of mortal flesh — and indeed the first night doesn't even require coffee, and the second presents just a 15% risk of getting a level of exhaustion. But as the DCs go up to impossible levels quickly, outside the bounds of 5E's math, everything goes downhill suddenly. Even with a streak of good rolls (early on, where they even matter), the eleventh night is certain death for even the most hearty hero.
Apparently sleep is significantly more important in D&D than in the real world; in 1965 a high school student stayed up for 11 days, with no apparent long-term harm. And there is no indication that this student was particularly proficient in Con saves, or even had class levels (beyond those available in high school). Scientists have monitored subjects kept awake for 8-10 days, also with no harm that couldn't be fixed by "one or two nights of recovery sleep" — which is much faster than 5E's slow recovery of one exhaustion level per night.
I find this somewhat disappointing: I know that D&D is not a good physics simulator, but I expected it to be pretty accurate on the subject of all-nighters.
In seriousness, I think the "and now you're dead!" thing is primarily there to make players take this seriously and not just never sleep except when spells need renewed. Personally, it's never come up in a game that I've run, but if it did, I would add "hourly con saves to stay awake" at level 4 and increase that to "every minute" at level 5, with further exhaustion only happening if some effect causes it or if you are forcibly kept awake.
The long rest mechanic is used for the purposes of restoring resources (spell slots, hit points, class feature uses, etc) to party members. Outside of this there is no reason, mechanically, for any character to take a long rest, let alone elves.
As you mention in your question, long rest and sleep are not the same thing -- sleeping is really a roleplay choice (or something that happens when you are affected by certain spells), whereas a long rest is use of game mechanics. Of course there may be penalties for not sleeping (your DM may cause exhaustion levels to increase, for example) but there are no penalties for not long resting (aside from the lack of recovery).
Ordinarily the two concepts are generally treated as the same or at least as happening at the same time ("We go to sleep and take a long rest,") but no, as far as your question is concerned and given the constraints you have provided, there is no need for an elf, or any other character for that matter, to long rest.
Best Answer
At least 8 short rests could fit in a long rest, presuming your DM is on board with the shenanigans
With reference to the basic rules, a short rest is defined like so:
And a long rest is defined thusly:
(Errata clarifies that you must sleep for at least six hours as part of a long rest, but that's not relevant to our problem - the overall durations are unaffected.)
Since a long rest requires a minimum of eight hours, and any given short rest requires a minimum of 1 hour, a character taking consecutive short rests while the rest of the party takes a long rest should be able to manage at least eight short rests in the period. If the rest of the party elects to continue the rest for a longer period, the character may be able to fit in another short rest for every hour their comrades continue lazing about. In most circumstances, though, one imagines that the adventuring party would not want to spend more time resting than they absolutely have to, so eight short rests would seem to be the standard.
Crawford Commentary (AKA Your DM Can Stop You)
Jeremy Crawford has weighed on this subject via twitter:
As he points out, the definition of a short rest requires a period of "at least 1 hour", and the intended way to play is that rests are defined narratively - so the DM is free to declare that a longer period of uninterrupted rest only constitutes a single short rest.
By cheesy RAW, the character can deliberately interrupt their current short rest after an hour passes simply by doing something strenuous, and then sitting down to rest again, so the period couldn't be defined as a single short rest. As a "strenuous activity" is not well-defined and could be no more than a few seconds of exercise, I assume the lost time devoted to forcing an end to your short rest is negligible when considering how many rests you can take and doesn't meaningfully change how many you can fit into a period.
However, Crawford goes on to state that the DM can decide how soon after finishing one rest the character must wait before they can start another, or how long the interrupting activity must actually be. If the DM is not on board with your shenanigans, this provides a Crawford-supported way to deny them - they can declare you just can't take consecutive short rests (we know the DM was always free to rule that way if they wanted, but now they have official backup, which may matter for some particularly argumentative players).
Actually, we're all elves
Sage Advice clarifies that, for elves:
So in the special case where the whole party (or at least the rest of the party) are elves, the short-resting character only gets away with 4 rests while the rest of the party meditates.