Any person, who reads Srimad Ramayana casually, and who views films and TV serials produced and developed by persons with half-baked knowledge, will say Yes the putra-kaam-eshthi yagna was performed however, a conscious and serious reading of Valmiki's Srimad Ramayana will tell a different tale.
King Dasharatha, a descendant of Ikshwaku kings, while dwelling in the city of Ayodhya, protected the Kosala kingdom. He had 3 main queens and many other queens. Unfortunately, he could not beget sons.
King Dasharatha decides to perform an elaborate Vedic ritual, called Aswamedha yajna, Horse Ritual, to beget children. He being a considerate king in taking conscience of the courtiers, he discusses this aspect with the Vedic scholars and ministers of his court, beforehand.
मम लालस्य मानस्य सुतार्थम् नास्ति वै सुखम् | तदर्थम् हयमेधेन यक्ष्ह्यामि इति मतिर् मम ||
"My mind is tumultuous without quietude for I have no sons... for that reason, I wish perform Aswametha, Vedic Horse Ritual... this is my thinking." (Bala Kanda 8th Sarga 8th Sloka)
तत् अहम् यष्टुम् इच्छमि शास्त्र दृष्टेन कर्मणा | कथम् प्राप्स्यामि अहम् कामम् बुद्धिः अत्र विचिन्त्यताम् ||
"Therefore, I contemplate to perform that ritual as enshrined in the scriptures and as a rite-oriented one as well... let this contemplation of mine be well thought of... and as to how my desire to beget sons will be fulfilled.." (Bala Kanda 8th Sarga 9th Sloka)
Everybody including Sage Vasistha agrees for this. On the advice of his minister Sumantra, Dasharatha proceeds to Anga kingdom and brings Sage Rishyasringa and his wife Shanta to his capital Ayodhya.
King Dasharatha requests the Sages to conduct the Vedic ritual for which the sages indent paraphernalia, which the ministers were ordered to supply forthwith.
So only Aswametha was thought of for begetting children by King Dasaradha and approved by everybody. And, the Aswamedha Yagna was completed by King Dasaratha.
The sloka 57 in Sarga 14 of Bala Kanda says like this:
Then that king(Dasaratha) is gladdened at heart for the successful achievement of the completion of that excellent Aswametha ritual that removes sin and that leads to heaven as well, and that which cannot be undertaken by many of the best kings.
It was only after completing the Aswametha, mention was made about Putrakamesti in the penultimate slokas of 14th Sarga of Bala Kanda.
The question is as to why the concept of Putrakamesti ritual was brought in when the completion of Aswametha ritual removes sin and children can be begotten.?
If we go through the subsequent Sargas, i.e., 15-17, we can understand that it was in these Sargas that
(i) the Devatas pray Lord Brahma and Lord Vishnu for elimination of Ravana, (ii) Lord Vishnu consents for taking an incarnation as Sri Rama, the son of King Dasaratha, (iii) After completion of Putrakamesti, a divine person appears and gives god-given golden vessel full with divine food for consumption by eligible wives of King Dasarath for begetting sons, (iv) King Dasaratha enters his Palace and gives the Divine food to Kousalya, Sumitra and Kikeyi, (v) Lord Brahma orders all Devatas to cause birth of vanaras out of their Energies to assist Sri Rama in elimination of Ravana.
However, in the Slokas 1 & 2 of Sarga 18 of Bala Kanda, the Sage Valmiki describes like this:
निर्वृत्ते तु क्रतौ तस्मिन् हयमेधे महात्मनः | प्रति गृह्य अमरा भागान् प्रतिजग्मुः यथा आगतम् ||
On the competition of the horse ritual of high-souled Dasharatha, the gods having received their portion of sacrificial oblation returned to their abodes as they have come.
समाप्त दीक्षा नियमः पत्नी गण समन्वितः | प्रविवेश पुरीम् राजा स भृत्य बल वाहनः ||
The king completing his consecratory vows held for horse ritual, entered Ayodhya city along with the company of his queens, servants, guards and vehicles.
Here, the Sage did not mention about Putrakamesti but mentions only about completion of Aswametha Yagna, horse ritual. The inference is that only the Aswamedha Yagna was contemplated, took up and completed successfully but Putrakamesti was not contemplated. It was not the intention of Sage Valmiki to project Sri Rama as the incarnation of Vishnu. It was a conscious insertion made by some scholars.
Were the Sargas 15-17 added later by somebody else for making the readers believe Sri Rama was an incarnation of Vishnu?
Am I correct?