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Based on the hadith mentioned that does not end only the way you related but also says something very specific which is ” there is no other way to make it up except this way…”
ففي الصحيحين عَنْ أَنَسِ بْنِ مَالِكٍ، – رضي الله تعالى عنه – أَنَّ رَسُولَ اللهِ صَلَّى اللهُ عَلَيْهِ وَسَلَّمَ، قَالَ: «مَنْ نَسِيَ صَلَاةً فَلْيُصَلِّهَا إِذَا ذَكَرَهَا، لَا كَفَّارَةَ لَهَا إِلَّا ذَلِكَ»
Anas reported God’s Messenger as saying, “If anyone forgets a stated prayer or oversleeps, expiation is made by observing it when he remembers it.” A version has “That is the only expiation for it.”
(Bukhari and Muslim.)
I think the Sheikh you heard is referring to the following hadith:
فَقَالَ النَّبِيُّ صلى الله عليه وسلم ” إِنَّهُ لاَ تَفْرِيطَ فِي النَّوْمِ إِنَّمَا التَّفْرِيطُ فِي الْيَقَظَةِ فَإِذَا سَهَا أَحَدُكُمْ عَنْ صَلاَةٍ فَلْيُصَلِّهَا حِينَ يَذْكُرُهَا وَمِنَ الْغَدِ لِلْوَقْتِ ” .
The Prophet (ﷺ) said: There is no negligence in sleep. The negligence is in wakefulness. If any of you forget to offer a prayer, he should offer it when he remembers it and the next day (he should perform it) at its proper time.
The hadith to the best of my knowledge encourages the person to remember to pray Asr at its time the next day and does not say to wait till the next day to make up for the missed prayer from the day before.
