All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah and that Muhammad is His slave and Messenger.
We have previously underlined in fataawa 82001 and 91480 that it is permissible to wipe over the socks for purification and explained the conditions for doing that. Even the Hanafis - whose school of Fiqh is dominant in India, Pakistan, and Asia in general - hold that it is permissible to wipe over the socks provided that certain conditions are met.
The Hanafi book Badaa'iʻ As-Sanaa'iʻ reads:
"It is allowed to wipe over the socks for purification as long as they are covered with leather or have leather soles according to the Hanafi scholars without difference of opinion. However, if the socks are not leather-soled or covered with leather and are thin, allowing water to penetrate and reach the skin, then it is not allowed to wipe over them for purification according to the scholarly consensus. If the socks are thick (but are not covered with leather or leather-soled), it is impermissible to wipe over them according to Abu Haneefah. However, Abu Yoosuf and Muhammad (the students and companions of Abu Haneefah) held that it is permissible to wipe over such socks. It has been narrated that Abu Haneefah changed his view on the subject shortly before his death as he wiped over the socks during his illness and told his visitors, 'I did what I used to tell people not to do.' Scholars relied on this incident to prove that he changed his view."
The Kuwaiti Encyclopedia of Fiqh reads:
"Socks are footwear made of wool, cotton, linen, or the like of fabrics. The majority of the scholars held that it is permissible to wipe over the socks for purification in two cases: First, when the socks are covered with leather because they are considered like the 'khuff' in this case. Second, when the socks have a leather sole. These two cases guarantee that water does not penetrate through them to reach the skin of the feet because water does not penetrate through leather.
Imaam Ahmad ibn Hanbal, Abu Yoosuf, and Muhammad (from the Hanafi School) held that it is permissible to wipe over the socks for purification provided that two conditions are met: Firstly, the socks have to be thick and cover the whole feet. Secondly, they could be walked in and are fixed on the feet by themselves without a need for laces or the like.
The Hanbalis did not stipulate the condition that the socks must have leather soles. They relied on the hadeeth narrated on the authority of Al-Mugheerah ibn Shuʻbah that says that, 'The Prophet wiped over the socks and sandals,' which proves that the socks did not have leather soles; one does not say, 'I wipe over khuffs and their soles.' They also relied on the fact that it has been reported that the Companions wiped over the socks and that there was no one who disagreed with them during their own time; therefore, it is considered that there was a consensus of the Companions on the permissibility of such an act... "
Imaam Abu Daawood wrote, "The following Companions wiped over the socks: ʻAli ibn Abi Taalib, Ibn Masʻood, Al-Baraa' ibn ʻAazib, Anas ibn Maalik, Abu Umaamah, Sahl ibn Saʻd, and 'Amr ibn Hurayth. It was also narrated that ʻUmar ibn Al-Khattaab and Ibn ʻAbbaas did that." [Sunan Abu Daawood]
Shaykhul-Islam Ibn Taymiyyah wrote:
"Imaam Ahmad compiled his book Al-Mas-h ʻala Al-Khuffayn (wiping over the khuffs) and cited the ahaadeeth and reports therein that indicate that the Prophet and the Companions wiped over the khuffs, socks, and turban. It was also reported that women wiped over their khimaar (head cover), like Umm Salamah, the wife of the Prophet and other women. Anas and Abu Moosa wiped over their turban-caps (taqiyyah). Pondering over this issue, one can realize the virtue of the scholars of hadeeth over the scholars of other fields of Islamic knowledge. Although the qiyaas (analogy) in this case is clear (between the khuffs and socks), some scholars of fiqh did not draw an analogy because they relied on the reports that reached them and chose not to draw the analogy (between the khuff and socks) out of prudence..."
It should be noted that none of the scholars, according to our knowledge, held that wiping over the khuffs or socks is a concession granted only during wartime or because of the scarcity of water, as the opinions that you cited suggest.
Allaah knows best.