If imo miyisrael is enough to allow one to have a position of srarah, why were bnei yisrael punished for comforting Agrippas (Sotah 41a-b) who was king with imo miyisrael (as Tosfos in many places asks)? (Note: see end of this post: if this Aggripas was Aggripas I, he was not imo miyisrael, so this question doesn't even begin, he was totally pasul and as Tosfos says, they only called him "brother" because avadim are obligated in certain mitzvos. If it was Aggripas II his mother was miyisrael, and this discussion follows this Agrippas. The lineage of both Agrippas I & II are discussed in that post.)
The basic answer is found in Tosfos in Yevamos 45b s.v. keivan, Ran Sandhedrin 36b s.v. chad (and Ritva Kiddushin 14a), who say that a king must have both parents be yisraelim, and not even one parent a ger, because of the special pasuk by melech: mikerev achicha.
The Ran suggests that Aggripas was a worse problem than father ger, mother bas yisrael, because Agrippas' father was an eved, and he therefore couldn't marry a kohen (see Minchas Chinuch 497 who brings Tosefta Sanhedrin 4:6 & Haghos Maimoni Melachim 1:4 that someone who may be appointed to srarah is equivalent to someone whose daughter may marry a kohen), but the son of a ger and a bas yisrael, a union from which their daughter may marry a kohen, could be melech. However, the Ran goes on to say even the son of a ger & bas yisrael couldn't be melech because you need both parents to be yisraelim.
Ran Sanhedrin 36b says Avshalom (who was mored b'malchus), whose mother was Maacha, a giores (formerly a yefas Toar) and his father was David Hamelech was qualified to be king. That is, his mother being a giores did not disqualify him from being a king because his father was a yisrael. The Kesef Mishna in Melachim 1:4 says this in regard to Rechavam ben Shlomo, (whose mother was a mitzris - bas Paroh) who did serve as king. We can push this off as a proof and say this does not show that if the father (or mother) was a yisrael that child may serve as king, because Avshalom and Rechavam were not appointed out of nowhere, their fathers were already king. So it could be that the din of yerusha in malchus and srarah overrides the requirement of both parents being yisraelim, which is necessary for an initial appointment, but not to appoint the son of someone who already has that srarah. In fact, this line of reasoning is brought by the Noda Biyehuda Kama, C"M 1, quoted by Minchas Chinuch 497 & 498 to answer why Rechavam was allowed to be king if his mother was a giores? Because his father was already king and Rechavam acquired it by yerusha, so he didn't need to have both parents miyisrael.
Ritva, Kiddushin 14a s.v. v'ha d'amrinan says that his Rebbi's Rebbi answered that when it says aviv miyisrael it's davka aviv, but only imo is not enough, but when it says imo miyisrael it's not davka, and could also be aviv. This second thing he says is like Tosfos in the previous post, where I wrote "Tosfos Yevamos 45b s.v. Keivan says that even if a yisrael married a giores, their son may be appointed; the child with aviv miyisrael has the same din as the son of a ger and a yisraelis, imo miyisrael." Tosfos means if imo miyisrael is enough, kal v'chomer aviv miyisrael is enough because he gets yichus mishpachas av from a father, which he doesn't get from his mother. So if imo or aviv miyisrael, he maybe could be a dayan, as we discussed in the previous post. If it's to be a melech, it must be aviv miyisrael davka, and imo moyisrael alone is not sufficient. What I'm not sure about in the Ritva, is that he says that you need aviv & imo miyisrael to be king, like the Tosfos & Ran at the beginning of this post, and then he says Rechavam ben Shlomo had a father from yisrael and mother a giores, and it was mutar because aviv miyisrael davka, as if you don't need both parents miyisrael to be king immediately after saying you do need both parents miyisrael as we see from Agrippas. Ritva uses this (Rechavam) to prove the point that when it says aviv miyisrael it's davka aviv, and Rechavam's father was Shlomo. Unless the point is aviv miyisrael is enough even for king, but Agrippas' father was an eved, so he couldn't be king. But the question remains why did Ritva just say you need both parent sfrom Yisrael in regard to Agrippas?
The bottom line is that this Ritva may be a different answer for how Rechavam was king - for melech, aviv miyisrael is enough.