Gotta remember that although some questioned his heritage, namely his brother Chagatai, Genghis himself stated Jochi was his son.
The point is that Genghis Khan himself stating that Jouchi was his son proves that Genghis Khan was a good man. Genghis Khan treating Jouchi real good all his life and even showing him and his sons some preference makes Genghis Khan even better.
But that did not prove that Jouchi was his biological son.
After wedding Temujin had sex with his young wife a couple of nights; after that she was stolen and the other guy had sex with her hereafter. In nine months Jouchi was born.
No one knows for sure who was a biological father of Jouchi, that is the point.
So saying everyone probably thought Jochi wasn't Genghis' son is wrong, but it would've defiantly caused a massive blow out in the Empire, but they may have supported his descendents, depending on what they offered and how the Kurlutai was handled.
I have to admit, that it is a very controversial matter.
But you see the Kuriltai is not gathering of the Chengizzids only, it is gathering of the Borjigin clan (which is much more numerous) and some of the Borjigins might think that Chengizz Khan stating that Jouchi was his son is a respected and even admirable deed. But that was his personal, family matter.
But the Great Universal Khaan is the head of the Borjigin Clan, first and foremost; and electing a man of questionable heritage as such might be... inappropriate.
And yet Jochi had enough support during his own fathers reign to try break off and quasi-rebel. So i think a far amount of people would've supported his descendants reign as Great Khan.
As I previously said all his life Chengizz Khan showed remarkable preference to Jouchi (probably to avoid any questions about his heritage; and probably feeling a personal guilt before him and his mother for letting this kidnapping happened).
This preference arose jealousy among other sons of Chengis Khan.
That envy was worsened by the fact, that Jouchi was the best general among all the Chengizzids/Borjigins; no doubt about it, he had a great experience of independent successful campaigning.
That is the reason and source of all these dirty rumors about Jouchi's being disloyal to Chengizz Khan. Majority of the modern historians think that this has nothing to do with reality.
I tend to agree with the modern researchers:
You see, you don't send the imperial army after your death to conquer a huge Ulus for the bloodline of your "treacherous disloyal" son, Jouchi. No, you don't. It is inconsistent with the Mongol mentality - the sons are held responsible for what their father did.