Stories of the Japanese 00618 [Early Kamakura Period] Efforts to Install a Shogun from the Imperial Court

On February 4, 1218 (the 6th year of Kenpo), Hojo Masako left Kamakura with her younger brother Tokifusa for a pilgrimage to Kumano. However, her true objective was to travel to Kyoto after completing the pilgrimage.
According to the Gukansho, Masako, while staying in Kyoto, met with Fujiwara no Kaneko (1155–1229), the wet nurse of the Retired Emperor Gotoba, and
requested, "As Sanetomo has no children, I would like you to send one of the Retired Emperor Gotoba's princes to the east to serve as his successor as shogun."
Upon hearing this request, Kaneko indicated that she would consider sending Prince Yorihito (1201–1264), whom she was raising as her foster son, to serve as shogun.
Now, this is a somewhat strange story. Sanetomo was only 25 years old and still had the potential to father children, yet it seems that not only Sanetomo himself but also Masako believed that
"Sanetomo will likely not have any children."
Was Sanetomo really that sickly?
Furthermore, of Yoriie's four sons, the eldest, Ichiman, and the third, Eijitsu, had been killed, but the second son, Kugyo, and the fourth son, Zenkyo, were still alive. It is a strange idea to go out of one's way to invite a shogun from the imperial family, who had no blood connection, when two of Yoritomo and Masako's grandsons were still alive. For the Hojo family, having killed Yoriie, was the option of letting his children inherit the shogunate already off the table?
The fact that negotiations for sending a prince were taking place at this point suggests that for Masako and Yoshitoki, a shogun of Minamoto blood was no longer necessary. During the Wada Conflict, Shogun Sanetomo failed to demonstrate any leadership, and everything was managed under the orders of the Hojo clan. Having gained confidence in their control over the gokenin through this incident, Masako and Yoshitoki may have thought to install a shogun who held only authority but no real power, allowing them to stand at the top in reality.
Now, installing a child of the Retired Emperor Gotoba as shogun was a choice with both pros and cons for both the Imperial Court and the Shogunate. From the Shogunate's perspective, they could rule the eastern provinces using the authority of the Retired Emperor Gotoba, and if necessary, they could use the prince as a hostage to confront the Imperial Court. On the other hand, from the perspective of the Retired Emperor Gotoba, as long as his child was the shogun, there was a possibility that he could control the Shogunate as he pleased.
It is difficult to say which outweighed the other, the benefits or the drawbacks, but the Retired Emperor Gotoba, upon hearing this idea from Kaneko, stated,
"I am thinking of sending a prince, but it is not something to be done immediately," and maintained a cautious attitude. It seems they decided to continue considering the matter for the time being.