It seems by now an unchallenged fact that ṭāġūt 'Idol' is a loanword from Classical Ethiopic ṭāʕōt 'id'. This word itself must certainly be a loanword from Aramaic ṭāʕū, ṭāʕūṯā 'error, idol', an abstract derivation (with the typically Aramaic abstract suffix -ū(t)) from the verb ṭʕā 'to go astray'.
I have an issue with this etymology. First, Arabic, as is readily visible, has a ġ in place of Classical Ethiopic ʕ. This is not an obvious way of borrowing this sound. Arabic simply has an ʕ, and if this were indeed a loanword from Classical Ethiopic, one would expect it to have been borrowed as **ṭāʕūt, not ṭāġūt. This issue has been recognized already by Nöldeke (2010) who I believe was the first to argue for the Ethiopic origin. He suggests that the word received its ġ from the verb طغى ṭaġā 'to transgress', a word which is, of course, cognate with the Aramaic ṭʕā.
This requires an extremely insightful etymological analysis from the Arabic speaker that borrowed it. When they borrowed the word '(false) idol', they would have had to have realized that this was an Aramaic -ūt, and that it would have to have something to do with 'transgression'. The meaning '(false) idol' is not evidently linked with 'transgression' anymore, nor is the -ūt suffix a productive suffix in Arabic. It therefore seems to me extremely unlikely that a speaker borrowing this word would have the incredible insight to connect ṭāʕōt with their own verb ṭaġā which just so happened to be cognate with the Aramaic word it derived from.
I'm of the opinion that the average speaker of Pre-Islamic Arabic probably was not a historical linguist, so I have a lot of trouble buying this narrative.
There is yet another problem. Arabic ṭaġā is usually thought to have two meaning 1. 'to transgress', 2. 'to overflow'. These are usually taken to be the same verbs, and meaning 1. can clearly be envisioned to develop from meaning 2. But taking these two verbs as the same verb appears to be a mistake that comes from an insufficient appreciation of Quranic Arabic orthography. In the meaning to 'transgress' ṭaġā is spelled طغى <tġy> (Q20:24) whereas in the meaning 'to overflow' it is spelled طغا <ṭġʔ> (Q69:11).
This is an important distinction. First, final ā written with a final yāʔ in fact conjugates differently from final ā verbs written with a final ʔalif. ṭaġaytu 'I transgressed' versus ṭaġawtu 'I overflowed'. Moreover, even in the 3sg.m. form (the only form attested in the Quran), in Quranic Arabic these words were certainly phonetically distinct.
It is clear from the Quranic rhyme that final "ā", written with a yāʔ were in fact originally pronounced with a vowel distinct from the final "ā" written with a final ʔalif. I argued for this in in my article on the Quranic triphthongs. In other words, we are dealing with two lexical items with two distinct pronunciations:
- /ṭaġē/ طغى 'to transgress'.
- /ṭaġā/ طغا 'to overflow'.
Now it's certainly likely that these two words do have a shared origin in some way. It is quite easy to see how 'to overflow' can take on a more abstract meaning 'to transgress' (the other direction, however seems difficult to motivate).
One is tempted to think, for this reason, that /ṭaġā/ is the original Arabic for, which never underwent a semantic development, and stuck to the rather mundane meaning 'to overflow'. It is at this point also important to note that it is typical of Arabic as opposed to, e.g. Aramaic and Hebrew, that it retained a distinction between these two final weak verb types. /ṭaġā/ coming ultimately from Proto-Arabic *ṭaġawa, while a verb like /ṭaġē/ would come from *ṭaġaya.
It seems likely that ṭaġē, then, is a loanword. in Both Aramaic and Hebrew the distinction between -awa final verbs and -aya final verbs has been lost, merging them all towards -aya-type verbs. Moreover, Aramaic and Hebrew both have undergone the semantic development from 'to overflow' to 'to transgress', and lost its original meaning completely. Considering the difference in semantics, when a speaker of Arabic was therefore confronted with the Aramaic or Hebrew word, it is not at all obvious that they would have connected it with their own (similar) word ṭaġā 'to overflow'.
At first, there might seem to be an issue here. Both Hebrew and Aramaic have merged ġ and ʕ to ʕ yielding ṭåʕå and ṭəʕā respectively. It would therefore seem that it is still unclear where Arabic got the ġ from. This, however, is not the problem that it seems. While neither Hebrew nor Aramaic writing have ever expressed the distinction between ʕ and ġ always simply using the <ʕ>, in both cases there is good evidence that this was purely an inadequacy of their script, rather than because they had merged these sounds already.
In Hebrew, the merger of ġ and ʕ must have been fairly late, as Hebrew names in the Septuagint still represent these sounds as contrastive (Steiner 2005). And as the same article by Steiner shows (pg. 235) Aramaic originally had a contrast between ʕ and ġ as well (as well as ḥ and x which in later Aramaic merge).
It moreover would not be the first time that an Aramaic word is borrowed into Arabic with a x or ġ where Aramaic today has ḥ or ʕ.
ṣibġah 'baptism', cf. Aram. ṣəḇaʕ 'to baptise'
nusxah 'copy', cf. Aram. nəsaḥ 'to copy'
xātam 'seal', cf. Aram. ḥāṯmā 'seal'
xardal 'mustard seed', cf. Aram. ḥarḏal 'mustard seed'
xāṭiʔah 'sin', cf. Aram. ḥṭīṯah 'id.' (although debatable, origin of the ʔ is difficult to understand)
xamr 'wine', cf. Aram. ḥamrā 'id.'
faxxār 'potter's clay', cf. Aram. paḥḥārā 'id.'
In light of these examples, it seems fairly obvious that *x and *ḥ had not yet merged to ḥ, and *ġ and *ʕ had not yet merged to ʕ in the Aramaic dialect that Arabic received this vocabulary from. As such, it seems much easier to argue, to me, that Quranic Arabic borrowed both ṭaġē 'to transgress' and ṭāġūt '(false) idol' directly from an Aramaic source, while the etymological cognate ṭaġā 'to overflow' was simply inherited, but was both phonetically and semantically distinct enough for these two verbs not to have merged. Gəʕəz would then have borrowed the word from either Arabic or Aramaic, there is no way to tell as Gəʕəz did lose the contrast between *ġ and *ʕ.
 Although I have trouble seeing why this would have to be an Aramaic loanword and isn't just an inherited Proto-Semitic word.