You are aware that the two sources you named give different names for imperial ancestor. First strike against him being truly related.
Yes, and this source is extremely suspect in that claim. If he had any of it he would not just name only his father and grandfather but all members of the line of Liu Zhen or Liu Fu. But they are not named. A normal family tree over three hundred years has roughly 12 generations. We know three from them, one, the source of the claim and the direct ancestors for him(Liu Bei). No sign of any connection. This makes the overall claim highly suspect. If you want to claim an ancestor, you normally have the line written out. Second strike for me against it actually being fact and not just a baseless claim or even just family legend. In every other circumstance someone claiming such a connection, would get laughed out of the room.
Then there comes the circumstances of when this all was announced. The point about his lineage was historically brought up when he himself became de facto emperor of Han or Shu-Han as it is better known. Because contrary to the King of Wu or the King of Wei, Liu Bei had no actual high rank title and could not claim some great connection to the Han court. He declared himself King of Hanzhong. Shu-Han was in dire need of a justification and there you have this obtuse connection. It is so vague, so old and considering Liu Zhen is said to have had 200 sons hard near impossible to disprove. Third strike.
Some of it would make me doubtful of his claim but all of it makes questions completely. Why it was never made a big deal of is actually simple. For the same reason you don't give true attention to claims about aliens or "fake news comments" or hoaxs. It is not worth it. The people followed Liu Bei for his massive charisma, abilities as a leader and not because he was the great^10 grandfather was a member of the imperial family. Like stated his claim is incredibly hard to disprove, but the burden of proof in this case should actually lay with the claimant and not someone else. If I claim to be a member of the Imperial family, I have to bring forth proof. If it is lacking, the claim is shit.
Furthermore, as I already said it is frankly speaking ridiculous to claim relations to another person from an ancestor 300 years ago. Considering how nowadays the biggest ancestry tree is reaching back 500 years and includes millions of people, it should become obvious how ambiguous or how much worth such a connection, if it truly existed, truly has/had.
I have a closer connection with my ancestors who fought in the Napoleonic War or some people in Israel then Liu Bei has with the Imperial Family, if his claim was true.