Well i hahve only two questions
1. Same words and same expression has been used for all companions of Prophet (s.a.w) participating in badr
Surat No 8 : Ayat No 11
اِذۡ یُغَشِّیۡکُمُ النُّعَاسَ اَمَنَۃً مِّنۡہُ وَ یُنَزِّلُ عَلَیۡکُمۡ مِّنَ السَّمَآءِ مَآءً لِّیُطَہِّرَکُمۡ بِہٖ وَ یُذۡہِبَ عَنۡکُمۡ رِجۡزَ الشَّیۡطٰنِ وَ لِیَرۡبِطَ عَلٰی قُلُوۡبِکُمۡ وَ یُثَبِّتَ بِہِ الۡاَقۡدَامَ ﴿ؕ۱۱﴾
[Remember] when He overwhelmed you with drowsiness [giving] security from Him and sent down upon you from the sky, rain by which to purify you and remove from you the evil [suggestions] of Satan and to make steadfast your hearts and plant firmly thereby your feet.
So do you agree that all conclusions you establish for all of those who you think33:33 applies also apply to all in Badr?
The intention of God sending water/tranquility from the sky was to establish them firmly and purify them, but the context, is that it works to keep them steadfast, not as a guarantee.
The talk of chosen households is different though. The Quran in different places mentions that God attributes purity to who he pleases and commands people not to do so themselves.
It's not that believers cannot become pure, it's that they are not allowed to claim it. The exception is those who God attributes purity to.
Ayat Tatheer is about who God through the Quran wishes to attribute purity to. When Zakariya (a) prays for a pure offspring, the meaning of pure here is a person made pure in the eyes of the people and is guaranteed to be without unclean blemish through out his life. Yahya (a) is purified in the eyes of the people and is without blemish.
A believer can become in a pure state, what they can't have is purity attributed to them by reputation of people.
The Quran praises some believers that organized the issue of accepting Mohammad (s) and his followers in Yathrib (later to become Madina) but said that God didn't make who did that known. And in fact, were it not for these people, Mecca would've been destroyed by God per Quran.
There is no doubt believers are praised in Quran. But Suratal Auli-Imran advises them to hold on to the rope of God and not turn on their backs in a tone that shows it's possible for them to do so.
The theme of chosen households in Quran, has it, that members of these chosen houses - are such that they are guaranteed safety and God knows they will not deviate from the path and any slips a Prophet every did in the past, God knew beforehand if they did slip they would return and never turn on their backs.
Ayat Tatheer is not a verse in isolation but has many verses about the chosen households in the past and that of Mohammad's (s).
The purification mentioned in the verse you are alluding to is them embracing the word of Taqwa, and it's to make them steadfast in hopes they do remain upon the path.
While the verses about chosen households are all in context, of manifesting and attributing purity by God to certain souls while no one can attribute purity to anyone really, but God.