I have significantly revised my views. However, I still do not think that Keiki/Yoshinobu was likely to have succeeded in modernizing Japan had he become Shogun in 1858 and instead would like to propose an alternative candidate.
There are two problems with a 1858 Yoshinobu. The first is that most accounts do not suggest that Yoshinobu was the sort of individual to commit himself to a revolutionary course of action. As mentioned by Maverick, most late Tokugawa leaders understood the necessity for some of the 1868 revolutionary changes that I mentioned above. Even Tokugawa Nariaki established a reverberatory furnace in Mito to cast cannon, working from a Dutch text. The most obvious important exception was the Emperor K[FONT=Times New Roman, serif]ō[/FONT]mei, who combined social conservatism with strong xenophobia. However, few if any of the early leaders seem to have envisaged anything on the scale of the post 1868 revolution.
There is some difficulty in using Shiba's historical novel, translated as "The Last Shogun", to establish an unbiased view of Yoshinobu despite Gibney's assertion that in the USA it would be regarded as straight history. Shiba seems to have written almost a defence lawyer's argument for Yoshinobu. A good example is how Shiba carefully avoids mentioning the fates of captured Tengu rebels from Mito and or their families (see “The Mito ideology: discourse, reform, and insurrection in late Tokugawa Japan” by J. Victor Koschmann for
the grizzly details especially page 155).
Other writers have seen Keiki as a Machiavellian figure (for example, The Last Samurai, pages 133 and 135) who turned exploiting the contradiction between the obvious need to agree deals with foreign powers and K[FONT=Times New Roman, serif]ō[/FONT]mei's fervent opposition to such deals into a career. This was the policy sometimes described as “verbal joi”, the idea that it would be possible to return to seclusion but not quite immediately (“The collapse of the Tokugawa bakufu, 1862-1868” by Conrad D. Totman, comments on
page 54 on Matsudaira Shungaku's 1863 efforts “The weeks at Kyoto demonstrated what so many at Edo had suspected: that a bakufu policy of imperial loyalism and verbal joi was untenable either as a way to restore Tokugawa leadership or to avoid a real collision abroad”). In any case, a look at Keiki's two main supporters, Tokugawa Nariaki and K[FONT=Times New Roman, serif]ō[/FONT]mei, does not suggest much hope for modernization.
The other problem for a 1858 Yoshinobu was that 1858 was already too late. Once Abe had written to the Daimyo and Hotta had visited Kyoto, the weakness of the bakufu was clear and all the proposed solutions involved the Emperor. For example, Ii Naosuke pursued a policy of union of the court and shogunate. However, as far as modernization was concerned, the Emperor K[FONT=Times New Roman, serif]ō[/FONT]mei was part of the problem rather than part of any solution (my already high opinion of the leaders of the Satcho alliance rose further when I encountered speculation that K[FONT=Times New Roman, serif]ō[/FONT]mei's death had been due to poison but, alas, Ravina states that his symptoms were consistent with a relapse from smallpox). Had a 1858 Yoshinobu attempted modernization, he would have had to face exactly the same opposition as Ii faced OTL although Ii Naosuke himself might have shown loyalty to a modernizing Shogun.
So who might have modernized Japan while maintaining the Shogunate? History is written by the winners and Il Naosuke has had few supporters. His Ansei Purge has been characterized as an attempt to shift from arguments, where the Shogunate was losing, to force, where they still held an advantage. It was already too late, force was met by force and Il himself was assassinated in March 1860. The Ansei Purge was not especially brutal and only eight people were actually executed. The most famous of these is Yoshida Shoin, described by Shiba as a teacher, which is true, but Shiba does not mention the detail that he had tried to organise the assassination of the Bakufu's representative at Kyoto. With perfect hindsight, we can see that it was foolish of Ii to kill an incompetent plotter who had very competent students such as Takasugi but it was hard for Ii to see so far. However, while finding the details of the Ansei Purge, I found a thread on Ii Naosuke at the excellent Samurai Archives Citadel, which included a letter written in October 1853 by Ii. It is immediately clear from
the letter that Ii was already an advocate of opening Japan in 1853 and it is tempting to make a lineage from Ii Naosuke to Katsu Kaishū to Sakamoto Ryoma to “Knowledge shall be sought through the world...”.
So what if Ii had been in Abe's position in 1853? There was no precedent for asking the Daimyo for advice (for example, Ieyasu seldom asked the Tozama Lords for their advice

). The Shogunate had established the Seclusion Policy without consulting the Emperor, so why should they need to ask him about reversing it. Ii could have simply opened Japan in 1853 and perhaps suppressed most dissent. Clearly, there would have been problems such as the Richardson Incident but probably fewer. There would have been economic problems arising from the need for more defence spending.There would have been problems over the Gold - Silver ratio as OTL. The Mito School would have continued to wield pens (or brushes) even if they did not use swords. There would even have been an Edo Earthquake in 1855. The Shogunate might well have survived all of these.
However, would it have been possible to transform Japanese society? The revolutionaries would perhaps have been men like Fukuzawa Yukichi (sent to America by Ii in 1859) or Itō Hirobumi (sent to Great Britain in 1863 as one of the Chōshū Five) as the Japanese who actually visited the West seem to have understood more of the transformation required. Without civil war and revolution we might expect much slower changes.