Full disclosure: the question is for a presentation I have to do. But the professor himself admitted that it would be difficult to find out. He was going to show us a film named "Tokugawa Yoshinobu" that would "give us a tip", but then he didn't and I can't find the movie. Maybe one of you can tell me where to find it?
Anyway, here is my hypothesis so far, based on what I've read. I didn't know much about the issue before last week when I started preparing, so forgive my potentially grave mistakes in understanding.
The first question is why his army was defeated in the first place, which I think I've figured out: parts of the Bakufu army were modernized, parts of it weren't, especially the seigniorial armys were fighting with traditional weapons and armor. The extent to which their opponents' armies were modernized was much greater, which was possible due to their differing economic strategies, allowing them to purchase more modern weapons etc... also some of the seigniorial armies defected to the other side.
There are also some elements of morale and fighting spirit to this, but I'm not sure I was able to understand this dimension in the short time I've had.
So the second question kind of presumes that, even though Yoshinobu lost the battle at Toba-Fushimi, his army was much stronger in terms of manpower and he would have potential foreign support at his side. But he surrendered fairly soon and easily, why? Or was it simply because his army was inferior? But the phrasing of the question suggests otherwise.
My guess: Previously he had already given over his nominal power to the emperor during the effort to modernize the governmental structure. His plan was to nevertheless maintain a high rank with in the planned parliamentary government, ensuring him a lot of influence while only giving up his position of Shogun. I understand that the civil war came to pass because other factions were not happy with him maintaining that much power (grossly oversimplified, of course). By surrendering early I assume he tried to save what could be saved to avoid the risk of being excluded from politics altogether or even exiled or killed. He wasn't that attached to the idea of keeping his position of the shogun anymore in the first place. But then again he didn't really wield any political power anymore after the Meiji Restauration, or did he? I read that at least his family line and the name of Tokugawa was allowed to remain intact.
So... please just let me know if my hypothesis is ridiculous or not and what you think could have been his reasons! I really apologize, I'm not knowledgeable about this topic at all and it seems to be a very complex following of events all influencing each other in many layers, so I am having quite some trouble with it. Also English is not my first language, so apologies for any unclear expressions and the like.
Edit: I'm not sure why my post has the "theatre" flair... and I don't know how to change it.
/u/AsiaExpert wrote a bit about this here.
I don't think your hypothesis is ridiculous, but this is one of the "we'll never know" things. He's dead and no one asked him.
However, given he had already resigned the position of Shogunate and Sacchou still pushed for war, he must have realized the high possibility that loosing the war in any way meant he would have no power afterwards (and he had none after the war). But he gave up so early in the war when he very much had the resources to keep fighting and perhaps even win. His navy was still intact and still the best in Japan. He had an arsenal and French support. And given what happened afterwards I don't think it's a stretch to say most of Eastern Japan would have followed him if he decided to carry on the war.
He also personally did not seem to dabble in politics in any way what-so-ever after the war, enjoying a life of leisure. All of this in contrast with his quite active leadership when he was Shogun.
So personally I believe he also did not want to raise arms against the Emperor.