Because otherwise, we historians wouldn’t have as fun comical things to write about!

No, really, a question like “Why did the Hojo not simply take the position of Shogun for themselves after the Minamoto shoguns were gone?”—or, for that matter, why didn’t the shogun become emperor , as had been asked a few weeks ago—is a bit reductionist: after all, it suggests that such a transaction is a matter between only two people, conveniently omitting the other millions of people out there who will have some kind of opinion on the affair (and at least a few hundred of these will be factored in directly into the decision-making process, especially those who command many other people with pointy sticks that you’d rather not have pointed at you).

Of course, we may engage in speculation what these reactions might have been, and we may attempt to use all the fun theories the past centuries of scholarship in the humanities produced to imbue some of the fragments of the past we know to an explanation (which takes some skill to prevent ending up an incomprehensible exercise in theory). But the truth of the matter is: we don’t really know “why,” because the people in question simply didn’t do it.

Invariably, any attempt of such explanation will almost certainly end up being a very elaborate way of saying “don’t fix what’s not broken”; since after all, people accepted that the Hōjō made the decisions in the name of the shogun, so, when reduced purely to the question of exercising power, becoming shogun themselves wouldn’t exactly be anything but a change in nomenclature. So, rationally speaking, what’s the point?

Anyway, that much for the, somewhat polemical, rhetoric :)

Let’s talk about what appears the state of your current knowledge—as far as I can infer that from the rest of your question—and what very real problems were (or: possibly were).

For starters, consensus in scholarship is that there was not one government during the Kamakura period, but two: the imperial court under rulership of the (retired) emperor, and the shogunate. Certainly, the shogunate was integrated in the symbolic framework of the imperial court (after all, the shogun was appointed by the emperor); still, we’re speaking of two parallel hierarchies.

For this reason, any discussion of the shogun-shikken-tokusō thematic is relatively independent of the imperial court, and I’m very eloquently going to omit this part of Kamakura politics (there’s probably a dozen threads on the whole court-shogunate thing somewhere on here anyway).

Now, let’s try and have at least some thematic structure: I’m first going to talk about the Hōjō and becoming shogun, then about the shikken / tokusō thing.

Yoritomo and Tokimasa, the Hōjō, and becoming shogun

Why did the Hōjō—after Yoritomo’s death, that would be old man Tokimasa—not become shogun?

Simple: he neither had the means nor the status to do so.

The Hōjō were only one of several powerful families (and they only had reached a semblance of parity with the real bigshots in the Kanto region, such as the Miura, due to their marriage ties to Yoritomo, and by slowly accruing economic means over a long time, including, and especially, after Yoritomo’s death). Naturally, an act as usurping the position of shogun (i.e., declaring yourself a superior to everyone else) at this point would have been ridiculously stupid—heck, Tokimasa’s attempt to replace Sanetomo with a candidate of his own was met with resistance by his own children, and others who had similar ideas in later years also weren’t exactly popular...

Ultimately, the Hōjō at this point in time were only special because Masako had married Yoritomo, and because Yoritomo’s successors were Masako’s children. It was therefore the much wiser option to reach a consensus with the other powerful parties present, lest they might have faced a couple hundred, or thousand, adversaries with their fifty-or-so retainers.

Although the idea to replace the shogun with somebody else ("just not me") was certainly present in the decades following Yoritomo’s death, it remains questionable if the idea of becoming shogun themselves ever was seriously considered by the Hōjō. Medieval society was very stratified in its ideas of status groups (Ger.: Stand), and which social position was appropriate for which rank, office, title, etc. The Hōjō, like everybody else, were part of this society, and as such had internalized their values—which his why its questionable whether the idea ever struck them in earnest.

Yoritomo, and his early successors, all were kugyō: members of the top aristocracy rivaling the status of the Fujiwara at court. The Hōjō (and all of their peers, for that matter) were not: they were low-ranked provincial officials who, by virtue of being related to Yoritomo through marriage made it to the mid-ranked, provincial governor-class of aristocracy. (Incidentally, this only happened after Yoritomo’s death, because it was Yoritomo’s policy to only elevate Minamoto, his kin, to this position.) So, there was a distinct gap in status between Yoritomo, his successors, and the Hōjō. This is quite important if we consider that the Kamakura shogunate, at its core, is a very elaborate super-bushidan blown way out of proportion, a hierarchical structure with one person being at the center of a network of multi-layered ties of personal vassalage (this person simply needs to be of higher status in society than everyone else because that's how hierarchies work).

We may argue that, in later decades of the Kamakura period, the Hōjō main branch could have attained kugyō status, had they chosen to pursue it. But taking a look at the historical facts, the Hōjō, throughout the entire Kamakura period, apparently had no real interest in advancing their status through rank and title beyond the fourth or fifth rank, remaining members of the zuryō (provincial governor) class until they met their end. This placed them quite comfortably at the top of the hierarchy of vassals to the shogun—they were the only non-Minamoto within the shogunate who were provincial governors—and kept them a few steps away from challenging the very fabric of medieval social hierarchy. Also, even if they had transcended this position and become kugyō, two issues remained:

For one, by then, they already had obtained the second-best thing you could get for legitimizing your rule: a son of the emperor as shogun. So, being kugyō would be, again, insufficient; and suffice it to say, becoming son of the emperor is probably not a very realistic proposition!

Secondly, they also lacked one more thing Yoritomo had: the narrative attached to noble heritage. They had no ancient history of being the heir to a famous warrior house, no pedigree, no history, no narrative that imbued them with some inherent charisma which made them inherently different from other Kantō warriors, and which would make it a natural choice for others to follow them. (Such a legitimizing narrative would only be created over the course of, and after, the Kamakura period.)

That being said…