But why vAlmIki rAmAyaNa, in its 24,000 verses, does not at least once, mention the name of ayOdhya as sAkEt or rAmA as the king of sAkEt or sAkEtapuri?
vAlmIki rAmAyaNa, in its 24,000 verses, mentions only once, the name of 'sAkEt'. But it is not to indicate ayOdhya.
It is in sundara kANda - the Book of Beauty (Volume 5), Chapter 1, verse 208, we find the word sAkEt.
tataH sa lambasya
vichitrakUTe nipapAta kUTE
saketak oddAlaka nALikere
hanumAn, the minister of forest-dwellers vAnaras visited rAvaNa's lanka, in search of Sita.
Immediately after leaping over the Sea, looking like a mountain of clouds, he alighted on the great mountain 'lamba'.
The Mount 'lamba' had a number of lower hill-cliffs.
The hills were full of 'kEtaka' (sa=with kEtak= kEtak trees) trees, uddAlaka trees and coconut trees.
This description is for the hills surrounding the lanka city and not ayOdhya.
Hence vAlmiki rAmAyaNa does not recognise the name 'sAkEta'.
Even rAmA's ayOdhya might have had gardens full of kEtaka trees.
Vol. 4 (Book of kishkindha - the city of forest-dwellers vAnaras), chapter 1, has 131 verses. The entire chapter has been devoted to describe the pampa lake and rAma's sexual passions and urges for his beloved SIta. Verse 4-1-81 makes a mention of the kEtaka tree.
Thus lanka and kishkindha had kEtaka trees.
nagara upavanam gatvaa
yathA sma ramate purA
tathaiva ramate siitaa
nirjaneSu vaneSv api
of Vol. 2 Book of ayOdhya says that SitA used to go to gardens (upavanam). But in Book of Childhood (bAla kAnDa or the Book of ayOdhya (ayOdhya kAnda), we cannot find any references to kEtaka trees, or gardens containing kEtaka trees.
Thus ayOdhya does not seem to fit into a name of 'sAkEt', though some Hindu scriptures other than vAlmiki rAmayaNa and mahAbhArata, mention ayOdhya and sAkEta as synonyms.