My question is: did 要 originally mean "to want to" and then get extended to mean "going to"? This would be similar to the history of "will" in English.
Yes. In the The grammaticalization of 要 and the future cycle from Archaic Chinese to Modern Mandarin, LaBarge (2016) states in the abstract that:
Similar to English will, 要 yāo/yào has developed new functional meanings apart from its earlier semantic meanings of Compulsion and Volition, including deontic and future time uses.
These all have more disambiguated disyllabic versions: 将要 (future), 须要 (the deontic meaning), 想要 (the full verb "want").
The character 要 itself is the original character for 腰, a woman pointing to her midriff = "waist". It was also used in the meaning "to arrange a meeting, to invite", which is now taken over by 邀. These meanings are attested in the earliest works of Classical Chinese literature, such as the 詩經 Shijing / Book of Odes.
Several centuries later, when we come down to the Analects, we see the meaning of "compulsion" take over (Analects 論語 Xian Wen 憲問, 14):
雖曰不要君,吾不信也。
Although it is said that (he) did not force his lord, I do not believe this.
This moves to the deontic meaning and the future meaning in the mid-Han dynasty. From the 漢書 Book of Han:
人生要死,何為苦心
Man, being born, must/will die. Why should this pain [my] heart?
Of course, this process happened a good few centuries before English's shift, but also before the shift in Romance languages too (present tense of reflex of HABERE + verb infinitive -> new synthetic future tense).
Interestingly in Modern Mandarin, 想 contrasts with 要, in a sort of indirectness as a way of showing politeness, when making requests. The author suspects that 要's directness is a result of its grammaticalisation.
Note that the non-cognate 欲 in many Min varieties, derived from 發 (Tu 2017), is similar in its semantic field of "want" through to "deontic need" to its grammaticalised use as a future, although the futurity remains at the inchoative or imminent level ("about to").