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.
It is 100 Hexes taking the Road and 99 taking the "shortcut" by the Lake, or 25 miles
Normal Pace: (24 miles/Day)(3 miles/Hour) 8.3 Hours
[16 total random encounter checks Up TO 2 encounters]
Fast Pace: (30 miles/Day)(4 miles/Hour) 6.25 Hours
[12 total random encounter checks UP TO 2 encounters] with -5 penalty on Passive wisdom checks
The distance traveled is possible, barring none of those encounters causes the party to slow down or break for recovery
If your overland travel is still too fast for your liking, consider adding thunderstorms, flash floods, or mudslides en route.
Best Answer
To begin with, we must understand that the 10-hour paralysis is a homebrewed effect, and such a condition was decisively not in view when the rules for resting were written, but we can try to surmise the interaction between the written rules and this homebrewed effect.
On the surface, the rules for long rests make no mention of the paralyzed or incapacitated conditions. So in the strictest RAW sense, there isn't a reason why you wouldn't benefit from a long rest.
But we can make some inferences from the description of long rests to rule the other way:
I would argue that suffering from a particularly potent paralysis poison for ten hours is going to fall somewhere in this category of "strenuous activity", but maybe that depends on how your DM wants the poison to paralyze you. Do you simply lose nerve function in your body and can't move, or does the poison violently lock up your muscles?
To determine if this paralysis falls under strenuous activity, you would have to ask the one who invented it.