How sound is the saying that it is possible to make up missed prayers by performing four rak'ahs on the last Friday night of Ramadan, without a middle tashahhud, and with the recitation of Surat Al-Qadr 15 times and Surat Al-Kawthar 15 times in each rak'ah?

The madhhab of the majority of scholars is that whoever misses the obligatory prayers, even intentionally, must make them up (qada'). This differs from Ibn Taymiyyah, who holds that making them up is not legislated and instead one should increase in supererogatory prayers (nawafil). The more authoritative opinion is that of the majority. What the questioner mentioned, that performing a prayer in the manner described absolves the one who missed prayers from the sin of what they previously left, is something for which we know no basis in Allah's Sharia.