Essentially, it appears what the scholar said is correct, and it is an opinion of early scholars i.e. that the verse states those who disbelieve in Islam disbelieve it in the presence of evidence and those who believe it believe because of evidence (if I understood you correctly).
However, another opinion of what the verse means exists which early scholars presented. In sha Allah, it seems all are valid and simply different ways of looking at it.
In the phrase "that those who perished would perish upon evidence and those who lived would live upon evidence" Allah is explaining one of his reasons for causing the battle of Badr.
Imam At-Tabari quotes Ibn Ishaq (the historian) about this verse:
١٦١٤٩ - حدثنا ابن حميد قال، حدثنا سلمة، عن ابن إسحاق: "ليهلك من هلك عن بينة"، [أي ليكفر من كفر بعد الحجة] ،(١٧) لما رأى من الآية والعبرة،(١٨) ويؤمن من آمن على مثل ذلك.
Ibn Ishaq said: "that those who perished would perish upon evidence" (8:42) [i.e. that those who disbelieve would disbelieve after clear argument]** of what they saw from the sign and lesson, and those who believed would believe upon the same. (Tafsir At-Tabari 68/13)
** Bracketed part is not from this narration Imam At-Tabari quotes but something added by the editor for clarification from the Seerah of Ibn Hisham.
(Translation is mine for all the quotes.)
An-Nahhas in his Ma'aani Al-Quran says:
قال أبو جعفر: قال ابنُ أبي إسحاق: جعل المهتدي بمنزلة الحيِّ، وجعل الضالَّ بمنزلة الهالك، قال: أي ليكفُرَ من كَفَرَ بعد الحجّة بما رأى من الآية والعبرة، ويؤمن من آمن على مثل ذلك.
Abu Ja'far said: Ibn Abi Ishaq said: (Allah) made (in the metaphor) the guided person like the living, and made the misguided person like the perished. He said: (The verse means) so that the one who disbelieves disbelieves after the clear argument of what he saw of the sign and lesson, and the one who believes believes upon the same. (Ma'aani Al-Quran 109/3)
(Note that round brackets are my additions for this and the next quote.)
In this interpretation, "those who perish" signifies "disbelievers" and "those who live" signifies "believers." This comparison of faith to life and disbelief to death exists elsewhere in the Quran (see 6:122)
This opinion appears to confirm what you heard this scholar say. Allah caused Badr to happen so that it would be a clear sign and lesson to everyone. So that after seeing this clear sign, disbelievers would disbelieve having seen the sign and believers would have stronger faith because of the sign.
Presumably, this would apply to all signs of Allah such that He would make clear to people what is true and what is false except that they would disbelieve out of complacency or denial. Allah says in the Quran:
We will show them Our signs in the horizons and within themselves until it becomes clear to them that it is the truth. But is it not sufficient concerning your Lord that He is, over all things, a Witness? (41:53)
Another way of understanding the verse exists. Imam At-Tabari says:
قال أبو جعفر: يقول تعالى ذكره: ولكن الله جمعهم هنالك، ليقضي أمرًا كان مفعولا= "ليهلك من هلك عن بينة".
ويعني بقوله: "ليهلك من هلك عن بينة"، ليموت من مات من خلقه،(١٥) عن حجة لله قد أثبتت له وقطعت عذره، وعبرة قد عاينها ورآها(١٦) = "ويحيا من حي عن بينة"، يقول: وليعيش من عاش منهم عن حجة لله قد أُثبتت له وظهرت لعينه فعلمها، جمعنا بينكم وبين عدوكم هنالك.
Abu Ja'far said: Allah says: Rather, Allah joined them there (at Badr) to finish a matter that had already been done (i.e. decreed by Allah): (the matter being) "that those who perished would perish upon evidence" [...]
And (Allah) means by his statement "that those who perished would perish upon evidence" so that whoever dies from his creation dies, with the argument of Allah made clear to him, with his excuse having been removed, and with having seen the lesson with his eyes. "and those who lived would live with evidence" (With this, Allah) says: So that whoever continues living from them lives with the argument of Allah having been made clear to him and having been shown to his eyes and mind. (For this) We brought you and your enemies together there. (Tafsir At-Tabari 68/13)
In the above interpretation, Imam At-Tabari takes the meaning of dead and living literally. According to this, Allah caused Badr so that the disbelievers who were supposed to die by decree would die (like Abu Jahl and his like). The argument of Allah had already been made clear to them, and thus they would have no excuse in the Hereafter. The argument of Allah had already been made clear to those who lived as well but Allah gave them more time.
The difference between the two opinions is that one states Allah caused Badr to give a clear sign to people after it.
The other states that Allah caused Badr to kill the disbelievers who needed to die. Signs had already come before, and those who died in Badr died having known the signs.
Both are correct. And Allah knows best.