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.
First off RAW gives the referee the authority to decide whether downtime activities are required.
Between adventures, the DM might ask you what your character is
doing during his or her downtime
In the PHB on page 187, we have a rule requiring the player to choose a lifestyle for their character and pay the appropriate expenses.
Between adventures, you choose a particular quality of life and pay
the cost of maintaining that lifestyle, as described in chapter 5.
Living a particular lifestyle doesn't have a huge effect on your
character, but your lifestyle can affect the way other individuals and
groups react to you. For example, when you lead an aristocratic
lifestyle, it might be easier for you to influence the nobles of the
city than if you live in poverty.
If your referee keeps strict track of time between adventure this expense can add up. If the income from adventuring doesn't cover it then the player may have to have their character do something in between adventures i.e. downtime activities to cover their expenses.
Now the lowest lifestyle is Wretched which cost nothing per day. However the description of Wretched is.
You live in inhumane conditions. With no place to call home, you
shelter wherever you can, sneaking into barns, huddling in old crates,
and relying on the goad graces of people better off than you. A
wretched lifestyle presents abundant dangers. Violence, disease, and
hunger follow you wherever you go. Other wretched people covet your
armor, weapons, and adventuring gear, which represent a fortune by
their standards. You are beneath the notice of most people.
While there no direct mechanical consequence of living a Wretched lifestyle, the part highlighted in bold give the referee ample authority per RAW to make the character's life a hell between adventures. The character would win at first but likely be worn down by the incessant attacks and interruptions.
So while engaging in downtime activities is not specifically required, it would not be smart not to at least do something to insure a decent lifestyle..
Best Answer
By RAW, it depends on what you are crafting; Yes, for some things.
As per the "Tool Proficiencies" section in Xanathar's Guide to Everything (starting on p. 78), with the correct tools many things can be crafted during rest periods. Each of these are specifically mentioned in RAW.
You can make basic alchemical materials like acid, alchemist's fire, antitoxin, or soap during a long rest.
Ditto a meal or a forged document takes a short rest.
Repairing metalwork, so smithing, can be done at 10 HP per hour but stipulates you need to have a hot enough flame to soften the metal, which in the real world means you need a forge wagon to do while afield, but here might be accomplished with magic.
Clothing can be created during a long rest, or repaired during a short one.
Arrows can be crafted during a short or long rest with woodcarver's tools.
Crafting a map is directly stated to something that can be done while traveling with cartographer's tools.
Painting and drawing is stated to be doable during rest periods.
Crafting also falls under the downtime activity rules (Xanathar's, p. 128), 50 gp value per 40 hrs of work, including some of the above as examples, implying some overlap and allowing us to compare time and progress.
The implication is clear anything you might reasonably be able to do with the tools on hand, you can do it, but other things are nigh impossible because the equipment or material is not portable. Notably masonry, glassblowing, and leatherworking lack any mention of rest activities which makes sense since these each require significant time and non-portable equipment or are themselves not portable (note in the real world harvesting a hide is very different from turning it into leather). Even smithing only allows for repairs with a big qualifier attached.
To use arrows as an example (XGE, "Woodcarver's Tools", p. 85), you can create 20 arrows during a long rest or 5 during a short rest, using downtime crafting 20 arrows takes about an hour (1 gp, 0.8 hours), so there is definitely an implication that crafting on the go takes a lot longer, about 4 times as long when it is possible at all. This makes sense since you would not be able to devote your full time and effort into it.
Yet something like an 50 gp antitoxin also takes a long rest (XGE, "Alchemical Crafting", p. 79), presumably because the cost reflects material cost not labor, which makes sense given the cost of raw materials given (50 gp per pound).
So it is up to the GM whether it is possible (do you have the necessary tools and raw material). How much you get done is not well-defined, but a good rule of thumb might be 1 gp worth of labor per long rest. Keep in mind that D&D is not supposed to be a high-fidelity life simulator; just because it doesn't talk about it does not mean you can't do it. Nowhere does it tell you you can bathe or how to construct a campfire either, yet few would argue you can't do these activities.