Just to clarify and be fair, the last statement in 33:33 switches from the feminine to the masculine, which means it is addressing all of Mohamed's household - including the males. What does not change is the subject matter, which is purification of the soul. There is still nothing there to facilitate the leap to claiming this proves anything about the 12 imams and succession.
Why is God talking to Mohammad's (s) wives? Is it due to their own merit or who they married to? I think the emphasis it's on the latter. So the subject in a way is about how wives of a holy chosen person by God should be extra careful. Do you disagree?
Now Ali (a), Fatima (a), Hassan (a) and Hussain (a), if special and chosen like Mohammad (s), can be brought here to emphasize their normal type family members (not special higher meaning) should be extra careful as well.
The line "God only desires.... " can be saying to the wives, you are married to a completely purified person from God and that's he is never at fault, and so be extra careful. He is completely and intensely pure, and so take care on how you address him, talk to him.
Now a similar tone happens:
لَا جُنَاحَ عَلَيْهِنَّ فِي آبَائِهِنَّ وَلَا أَبْنَائِهِنَّ وَلَا إِخْوَانِهِنَّ وَلَا أَبْنَاءِ إِخْوَانِهِنَّ وَلَا أَبْنَاءِ أَخَوَاتِهِنَّ وَلَا نِسَائِهِنَّ وَلَا مَا مَلَكَتْ أَيْمَانُهُنَّ ۗ وَاتَّقِينَ اللَّهَ ۚ إِنَّ اللَّهَ كَانَ عَلَىٰ كُلِّ شَيْءٍ شَهِيدًا | There is no sin on them [in socializing freely] with their fathers, or their sons, or their brothers, or their brothers’ sons, or the sons of their sisters, or their own womenfolk, or what their oaths hold authority over
And Be wary of Allah. Indeed Allah is witness to all things. | Al-Ahzaab : 55
You may not know, but the bold is feminine plural and hence addressed at the wives of the Prophet (s). The next verse is:
إِنَّ اللَّهَ وَمَلَائِكَتَهُ يُصَلُّونَ عَلَى النَّبِيِّ ۚ يَا أَيُّهَا الَّذِينَ آمَنُوا صَلُّوا عَلَيْهِ وَسَلِّمُوا تَسْلِيمًا | Indeed Allah and His angels bless the Prophet; O you who have faith! Invoke blessings on him and invoke Peace upon him in a worthy manner. | Al-Ahzaab : 56
So we see, that although God is addressing believers in 33:56, there also a special application to the wives 33:55 in that they have to be very careful regarding God because of who they are married to.
Now what is interesting is that when 33:56 was revealed, even according to Sunni hadiths, the Prophet (s) taught to bless his family along with himself, in which all Muslims do in Salah. That is we bless Mohammad (s) and the family of Mohammad (s).
33:33 can be also saying to the wives you are married to someone who God desires to keep nothing away from blessings, but rather, all he wishes to keep away from his holy soul, is uncleanness and that God purifies him a complete purification. The only thing is that Ali (a), Fatima (a), Hassan (a) and Hussain (a) are included in that exaltation by God over others.
Now which one of these is the proper meaning. Sunni and Shiite hadiths both have 33:33 (the 2nd part) revealed for Ali (a), Fatima (a), Hassan (a) and Hussain (a). Notice, that Zainab (a) also a daughter of Ali (a) and Fatima (a) was never included in the event despite being born at that time.
Why is that the case?
Same with Mubahila event, Zainab (a) who is honorable lady and a hero, is not taken, to that event. This is because she is a high believer, she is not part of the chosen household, she is from the household much like Salman (a) is from Ahlulbayt (a) but not an actual member of the holy household.
There's more to it. Back up a little where else do wives and relatives (chosen) appear in this Surah?
Go to verse 33:6, we see the Prophet (s) is emphasized to have a priority right of authority over believers then they over themselves, and then it mentions "and his wives are their mothers", so we see wives are emphasized not due to their own merit, but who they are married to. But the verse then says "and the possessors of blood/family relationship, some of their are AWLA then others....".
Now Sunnis take that and say this about believers blood/family, but the context is clear, it's about Welayat of the Prophet, and so the Awliyakum here is about that Welayat mentioned in start of the verse, and so to not fragment the verses into many irrelevant to each other subjects. The proper way to see it is that it's saying, and his holy family also have this authority with some of them preceding others in taking it (ie. Ali (a) first, then Hassan (a), then Hussain (a), etc)....
So a better translation of what it says is: "only that you should towards your Masters/Authorities (ie. those who have more authority over believers then themselves) goodness indeed this written in the book".
The next verse then emphasizes on the covenant of all Prophets (a) - and the next verse "that truthful maybe asked about their truthfulness" and disbelievers and hypocrites to be punished.
Now 33:33 in light of that, can be going back to 33:6. The wives are emphasized as our mothers, but are they special on their own merit or are they emphasized because they are married to Prophet (s)? It seems the way they are told to decide between Dunya or God/his Messenger/Next world, is that, they are not chosen like Sarah (a) (wife of Ibrahim (a)). On the other hand, the "Ulul-Arham" (relatives of the Prophet (s)) meant in 33:6, are chosen to be of the household of Prophet (s) because of the pre-world covenant and they were chosen by God. They are special.
Now in the Sunnah (hadiths) we find:
33:6 is about leadership of the family of Mohammad (s), and they are Ulul-Arham (I will share hadiths).
33:33 (the 2nd part) is revealed about five souls at that time, and Zainab (a) who is a daughter of Ali (a) and Fatima (a) is not included.
33:56 - the proper way to bless Mohammad (s) is to bless his family with him.
33:33 and 33:55-56 are similar, in that they tell the wives that Prophet (s) is extra special.
Another point since there is two ways to recite 33:33, we have to revert to what is more clear in Quran. The question, per Quran, is there a holy chosen family. If you recall the reward verses 42:23, 25:57, etc, it's obvious the family of Mohammad (s) is chosen.
If you recall and connect 4:54 with 4:59, it's obvious the family of Mohammad (s) are the leaders of authority from God in this nation, just like the holy family of Abraham (a) were given a great authority, and so it's telling us not to envy our leaders and the holy family of our time, and how irrational it is to believe in past chosen families like the family of Abraham (a) and honor them, but deny it the current household of guidance.
So this way, we can say 33:33 in itself has many possible ways of reading. However, when we revert to clear parts of Quran that show the household of Mohammad (s) to be special (like 42:23), 33:33 (the latter part) then is to be read with that kept in mind.
I wanted to also discuss the double purification. Per Quran, God attributes purity to who he pleases (shows who is pure) and it's not for people to do so. But regarding Mariam (a), we don't see this double emphasis. Why, because there is levels of purity. The Prophets (s) themselves differ in rank with respect to how holy and intensely pure they are.
There is a line in Saheefa Sajjadiya to bless Angels that are chosen by God (the du'a about Angels) with a blessing that will increase their purity on top of their purity.
Double word in Arabic, like "daka daka", means it takes a very intense and full out and even extra form of that.
No matter who they are married to, I don't think God wishes to raise wives of Prophet (s) above the purity of Mariam (a) and Prophets (a) who he has chosen above them.
We see in Quran, believers are purified, the word of Taqwa or tranquility they receive it. The Satanic filth he removes. And that God wishes through Salah and rituals of connection to God, to purify believers.
But he never uses this double purification expression. We find this double meaning, for day of judgment, "shaking a shaking", meaning there is no shaking like it. It's the most severest form of it.
So Mohammad's (s) family have most intense form of purification.
This is something you are not familiar because double words in English are said to be grammar wise wrong or it's not considered eloquent to double emphasize. However, in Arabic, the usage has this meaning. It's not just a complete form of it, it's the ultimate of it.
So Ahlulbayt (a) of the ultimate form of purity. We can be pure, but God doesn't intend we reach higher status in purity then Mariam (a), Jesus (a) etc.
Anyways there is a lot covered in this post. Let me know what you think of each point and if you have to do separate posts for each point, that's fine. It was a bit long I apologize.