Since all of this happens inside a house or home. Thamizhs referred to their house as அகம். Over time வீடு replaced அகம் as the word thamizhs used in order to refer to their "home". So usage of "என் அகம்" and "என் அகத்திற்கு" became "என் வீடு".Obviously I obtained this knowledge by asking a few thamizh literature enthusiasts. None of them really knew the origins of வீடு. Some thought that houses in the olden days came from ஓடு and so வீடு was derivative of that. But no one was sure. However, even though வீடு became popular over time, many continued using அகம் . But over time அகம் narrowed down in usage to members of specific caste. Also, over time வீடு became architected by madras thamizh as ஊடு. And brahmins began to shorten and mangle அகம் to ஆம் and அகத்திற்கு to ஆத்திற்கு. There is a small minority theory that अहम् from अहम् ब्रह्मास्मि fame was the origin of ஆம். Because in the context of अहम् ब्रह्मास्मि few do argue that the phrase means "ब्रह्म resides inside of 'me'" as opposed to "'I' am ब्रह्म". But except for 1 or 2 people who gave this as the reason the brahminical usage of ஆம் to refer to 'house' seems to originate from அகம்.
Coincidentally, ஆம் has competition in terms of bad usage. ஆறு means river. ஆற்றிர்கு is usage of ஆறு as in ஆற்றிர்கு செல்கிறேன். All thamizhs regardless of caste have consistently murdered this word to pronounce it as ஆத்திற்கு செல்கிறேன். While many often makes fun of people who use ஆத்திற்கு instead of அகத்திற்கு (as in 'enge thamarai Bharani'ya? Kaveri'a? )the reality is there no such word called ஆத்திற்கு. There is आम though. And its very tasty. But its a வட மொழி சொல்.
Disclaimer: All this is kelvi gnaanam. The purpose of posting it is to see if someone has a better explanation
P.S: Inspired by origin of 'summa'