Rama's father Maharaja Dasaratha had 3 Queens and 350 other wives. On the other hand, Rama is Ekapatni Vrata, the man who wedded only once. Sitadevi is his only wife.
This is illustrated many times in the Ramayana, prominently in the episode of Surpanakha, where he denies her proposal saying he is Krta Darah, meaning one who has already married - this implies his Ekapatni Vrata.
Moreover, the chapter linked in the question contains a monologue by Manthara, who was at that time in the process of poisoning Kaikeyi's mind. So her opinion of Rama is not really reliable. Even so, the words she speaks are to be noted:
Hrshtah Khalu Bhavishyanti Ramasya Paramah Striyah|
Aprahishtah Bhavishyanti Snushaas Te Bharata-kshaye||
Here, she says, Bhavishyanti, which means, "Will become". So the verse, if put into the following anvaya:
Hrshtah, Paramah Striyah Ramasya Khalu Bhavishyanti,
it means, being delighted (Hrshtah), the princesses (Paramah Striyah - literally great women) will become Rama's (wives).
So it does refer to marriages of Rama after he becomes the King.
Furthermore, in the Uttara Kanda, the priests of Rama advise him to marry someone else, so that he would be able to perform the Ashvamedha Yajna, which would not have been said if Rama had multiple wives. Even at this point, Rama refuses to marry and instead seats beside him a golden image of Sita.
Also, the idea that what we follow today is Tulasidasa's Ramayana, and not Valmiki's, is another fragment of Mr CR Sreenivasa Iyengar's wild imagination.
The Ramayana as we know it is definitely influenced by Tulasidasa, in places like the meeting of Rama and Sita in Mithila Gardens and Lakshmana's Rekha, but on the whole, most versions remain faithful to Valmiki.
(And by most versions, I include only versions entitled 'Ramayana', not 'Ramacharitamanas' or 'Kamba Ramayana' or any others.)