Kaikeyi was not thinking straight in Valmiki Ramayana. She was daughter of Ashwapati, ruler of Kaikeyas and sister of Yuddhajita.
Kaikeya Kingdom in Valmiki Ramayana is mentioned to be in Bahlikas (Bactria). In Mahajanpada period, Kaikeya Kingdom had come to modern day Pakistani Punjab. But in Valmiki Ramayana, the messengers of Vashishtha, crossed Sudama Mountains in Bahlika (Balkha) and then went straight on the road to Rajgriha (Royal house of Kaikeya Kingdom). So, most likely Kaikeyi was a Bactrian Princess. The name of her father Ashwapati which means Lord of Horses even makes this point stronger.
Kaikeyi was very beautiful and Dashrath doted on her. She had a son named Bharat. Due to extreme affection of Dashrath, she used to not treat his other wives properly. This doesn't mean she went to verbally insult them but her voice mattered way more than other queens including Kaushalya. Even Kaikeyi’s servants had more say than other Queens.
When Manthara told Kaikeyi that Shri Ram was to become new King, the initial reaction of Kaikeyi was of joy not any hatred because Ram had only respected Kaikeyi like a mother. The personality of Shri Ram was as such every person in the Kingdom loved him.
When Manthara’s repeated manipulations didn't bear any fruits, she played her master card and said Kaushalya would treat her the same way Kaikeyi has treated others. That triggered her anxiety.
Kaikeyi not wanting to be treated inferior to Rajmata which Kaushalya would become , she demanded her infamous boons which brought destruction on her life. She lost love of Dashrath and also her husband’s life. She lost respect of her son Bharat and the whole Kingdom.
Her voice and opinions didn't matter anymore even when Shri Ram was in exile.
Manthara was just a servant, she was thinking as benefactor of her mistress but Kaikeyi was the one with a education, power and responsibilities. A real queen is someone who listens to everyone but takes her own decisions.
It's Shri Ram's grace that Kaikeyi was not punished not only because she was a princess and protected by her Kingdom because Shri Ram would never hurt someone who was once genuinely loved and respected by her father and him as his mother.
Shri Ram was not some Kaikeyi worshipper in Valmiki Ramayana unlike TV serials, his own mother Kaushalya mattered to him the most but he didn't let his brain filled with hatred for Kaikeyi.