Correction of your statement and explanation of the terms
Certainly the qur'an is the first choice to understand the qur'an and its grammar. One could say Arabic grammar needed a standard book or source to be finalised and fixed and this "standard work" was the qur'an in form of al-Moshaf al-Imam which excluded a lot of deviations or variations of the spoken Arabic. As a matter of fact the first books on Arabic grammar were only compiled during the reign of the Umayyads.
And the qur'an actually clearly explains or maybe its better to say hints that سوف is generally used to express a far future while س is generally for a close or nearer future unlike you've claimed in your post which actually is the opposite of what was mentioned in the shared video which correctly explained it.
سوف is also related to تسويف which means (source almaany)
- Avoiding committing oneself so as to to gain time
It describes the case of people who intend to do something but are not upstanding and firm in their intention and keep putting that intention off: So they say I'll do it tomorrow, and repeat it the next day until they either really do what they intended or die without fulfilling that intention.
Therefore it is said that the meaning given by سوف provides an extension so it extends the time or time span for an action to be performed. However in the context of a warning or promise of Allah we must be aware that it will come true at latest in the hereafter.
While سوف may imply that something might happen (or may not happen if for example the addressed person acted otherwise), سـ implies a certainty that something will happen.
In other words:
سوف = + سـ
meaning سوف covers what سـ covers and can cover more. But the statement the later is related to is more certain to occur than the first.
Maybe a kind of additional rule can be wherever both apply and there's a hint of a doubt it is best to use سوف. While سـ needs a certainty.
Examples from the qur'an on the use of سوف
Back to the qur'anic evidences for my claim above you may notice that most of the verses in which سوف is used are addressing something that will happen in the hereafter which is in the far future as there's at least death in between for example:
Indeed, those who disbelieve in Our verses - We will drive them into a Fire. ... (4:56)
إِنَّ الَّذِينَ كَفَرُوا بِآيَاتِنَا سَوْفَ نُصْلِيهِمْ نَارًا ....
But they who believe in Allah and His messengers and do not discriminate between any of them - to those He is going to give their rewards. And ever is Allah Forgiving and Merciful. ... (4:152)
وَالَّذِينَ آمَنُوا بِاللَّهِ وَرُسُلِهِ وَلَمْ يُفَرِّقُوا بَيْنَ أَحَدٍ مِّنْهُمْ أُولَـٰئِكَ سَوْفَ يُؤْتِيهِمْ أُجُورَهُمْ ۗ وَكَانَ اللَّهُ غَفُورًا رَّحِيمًا
And O my people, work according to your position; indeed, I am working. You are going to know to whom will come a punishment that will disgrace him and who is a liar. So watch; indeed, I am with you a watcher, [awaiting the outcome]." (11:93)
وَيَا قَوْمِ اعْمَلُوا عَلَىٰ مَكَانَتِكُمْ إِنِّي عَامِلٌ ۖ سَوْفَ تَعْلَمُونَ مَن يَأْتِيهِ عَذَابٌ يُخْزِيهِ وَمَنْ هُوَ كَاذِبٌ ۖ وَارْتَقِبُوا إِنِّي مَعَكُمْ رَقِيبٌ
See also (4:30), (4:74), (4:114), (4:146), (5:16), (6:5), ...
And in the example discussed in the video:
No! You are going to know. (3) Then no! You are going to know. (102:3-4)
كَلَّا سَوْفَ تَعْلَمُونَ ﴿٣ ﴾
ثُمَّ كَلَّا سَوْفَ تَعْلَمُونَ
Here the repetition is a confirmation and assertion.
Basically the only incidents where سوف was not used for expressing a warning or promise of Allah in the qur'an are the promise of Yusuf():
He said, "I will ask forgiveness for you from my Lord. Indeed, it is He who is the Forgiving, the Merciful." (12:98)
قَالَ سَوْفَ أَسْتَغْفِرُ لَكُمْ رَبِّي ۖ إِنَّهُ هُوَ الْغَفُورُ الرَّحِيمُ
I have not checked any tafsir to comment with more certainty, but I'd assume that this promise came in this version to express that even if Yusuf from his site may have forgiven them maybe they committed sins that Allah only can forgive. But Allah knows best.
And that of Pharao:
Said Pharaoh, "You believed in him before I gave you permission. Indeed, this is a conspiracy which you conspired in the city to expel therefrom its people. But you are going to know.
قَالَ فِرْعَوْنُ آمَنتُم بِهِ قَبْلَ أَنْ آذَنَ لَكُمْ ۖ إِنَّ هَـٰذَا لَمَكْرٌ مَّكَرْتُمُوهُ فِي الْمَدِينَةِ لِتُخْرِجُوا مِنْهَا أَهْلَهَا ۖ فَسَوْفَ تَعْلَمُونَ
Here the explanation one can use is that what happened is still narrated until today and it is a human who made the warning. And Allah knows best.
And that his effort is going to be seen - (53:40)
وَأَنَّ سَعْيَهُ سَوْفَ يُرَىٰ
here the next verse completes the meaning, what men will do will be seen in this life and recompensed in the hereafter.
In 5:54 we find a warning and confirmation of something that will certainly happen in this life. But as Allah addressed the believers the meaning can be extended to believers of any epoch, which again confirms the use of سوف. This also applies to the statement of Dhi-Al-Qarnayn in (18:87).
In verse (6:67) a realisation in this life is as possible as one in the hereafter.
There are also verses where سوف is expressing something that some of the people who were initially addressed with may witness "soon" for example (9:28), but the order is remaining.
The remaining 5-10 verses in which سوف was used in the qur'an are covered in above explanations and comments.
Some use cases of the future expressed by verbs beginning with سـ in the qur'an
Here I'll just show a handful of examples as there are far too many to address all of them, so I hope ypu'll forgive my cherry picking:
The foolish among the people will say, "What has turned them away from their qiblah, which they used to face?" ... (2:142)
سَيَقُولُ السُّفَهَاءُ مِنَ النَّاسِ مَا وَلَّاهُمْ عَن قِبْلَتِهِمُ الَّتِي كَانُوا عَلَيْهَا
In the nearest land. But they, after their defeat, will overcome. (3) Within three to nine years. To Allah belongs the command before and after. And that day the believers will rejoice (30:3-4)
فِي أَدْنَى الْأَرْضِ وَهُمْ مِنْ بَعْدِ غَلَبِهِمْ سَيَغْلِبُونَ * فِي بِضْعِ سِنِينَ لِلَّهِ الْأَمْرُ مِنْ قَبْلُ وَمِنْ بَعْدُ وَيَوْمَئِذٍ يَفْرَحُ الْمُؤْمِنُونَ
Here Allah shows that this will happen in a few years.
About what are they asking one another? (1) About the great news - (2)
That over which they are in disagreement. (3) No! They are going to know. (4) Then, no! They are going to know.
كَلَّا سَيَعْلَمُونَ (4) ثُمَّ كَلَّا سَيَعْلَمُونَ
I've quoted here in English more context than in Arabic. This verse actually would jeopardise my claim above because from the perspective of the disbelievers the future in which they will "know" the truth is far away in the hereafter, however from the perspective of those who believe or will believe they will be believing in the day of judgement from the moment they've faced the truth in this live.
Further a-Dahhak has been reported to read these verses explaining as follows:
No! They -the disbelievers- are going to know (4) Then, no! They -the believers- are going to know.
Nevertheless these verses in first place express the certainty of the day of judgement.