As someone who has spent a substantial amount of time this year researching the elusive Hōjō genealogy, I can say the following:

It is very common for a warrior, or any similar kind of lesser aristocracy, to be hardly more than a name in genealogies—especially when we are talking about the Heian and the first decades of the Kamakura period, where documentary evidence is quite a bit scarcer than it becomes ca. from the mid-13th century onward. All too often, we are stuck with a few words gleamed from genealogies, which are known to not be the most accurate kind of source. That being said, this is what I got:

In the massive 14th-century compilation of genealogies, the Sonpi bunmyaku, the only information given about Minamoto no Yoshitoki are: his azana (call name) was Mutsu no Gorō 陸奥五郎, his position at court sahyōe no suke 左兵衛佐 (a Second-Ranked Officer of the Right Watch), and he was also referred to by the gō (title) Ishikawa 石川 (1). Another section of the Sonpi bunmyaku identifies him as Mutsu no Rokurō 陸奥六郎 instead (2).

Other, less extensive records of this kind say the following: the “Shoke daikeizu” identifies him as Mutsu no Rokurō, sahyōe no suke, as does the “Keizu san’yō” (3); in sum, there is more indication of having been the sixth, not the fifth son. The “Seiwa Genji keizu” in the Gunsho ruijū doesn’t offer anything beyond a name and the court office of uhyōe no suke 左兵衛佐 (Second-Ranked Officer of the Left Watch)—confusing the left and right positions of court offices is a very common mistake in genealogies (4).

A document issued at the occasion of the in-no-hajime 院始 of Fujiwara no Akirako (also incorrectly known as Shōshi), dated on the 12th month, 20th day, mentions Minamoto no Yoshitoki as one of the four new appointees to hyōe no suke (5). (An in-hajime is the establishment of a retired emperor, empress, imperial mother's household administration in the wake of receiving their title.) The document makes no mention of the year of issuance, but we know that she received her ingō, Taikenmon-in, in 1124, which should settle this. Taikenmon-in is famously the mother of emperors Sutoku and Goshirakawa.

This is the only documentary evidence making any mention of Yoshitoki. It is barely anything—it merely offers a source for the information given in genealogies—but we can infer a few things:

1. With certainty, Yoshitoki, like his father, was based in the capital for a significant amount of time, else he would not have been eligible for appointment to hyōe no suke (at the time, holding court office still required presence in the capital).
2. If I infer from other biographies, the appointment to a hyōe no suke position in the wake of the establishment of a join's (a member of the imperial family holding an in-title) household administration leaves room to speculate that Yoshitoki was quite likely in a clientele relationship with Taikenmon-in, which may also have meant that he might have held some lower-ranked office in her household administration at some point; it may also indicate that, perhaps, Ishikawa Estate was part of her property at the time, although there is apparently no evidence about Ishikawa Estate before 1152 out there. But all of this is just within the realm of possibility.

Otherwise, all you get is that Ishikawa Estate is said to have been created by Minamoto no Yoshiie, and it is assumed that, after Yoshiie’s death, Yoshitoki probably became the estate manager, which is, again, merely inferred by his son Yoshimoto being known to hold the position (6); likely, the Sonpi bunmyaku included the mention of Ishikawa due to this connection.

But to lastly explicitly answer your title question (although it should have become apparent by now): yes, he apparently existed, since otherwise, how could there be a document appointing him to office at the imperial court? And, as I initially mentioned, that is more than most others get.

Notes:

(1) Sonpi bunmyaku, vol. 3, pg. 237.

(2) Sonpi bunmyaku, vol. 3, pg. 304.

(3) As per excerpts printed in Dai Nihon shiryō, vol. 3.8, pg. 735.

(4) In Zoku Gunsho ruijū, vol. 5 jō: Keizubu, pg. 266.

(5) Heian ibun, doc. 3721.

(6) Nishioka Toranosuke, “Ishikawa no shō no seiritsu to Kawachi Genji no hatten,” in Shien 5:6 (1931), pg. 376. Digitally available via https://doi.org/10.14992/00000654 .