According to the family tree, there must be a blood connection.
The branch of Liu Xiu's ancestors started from Liu Fa, King of Changsha. Liu Fa's father was Liu Bang's grandson, Emperor Jing Liu Qi of Han Dynasty (Liu Fa's brother was the King Jing of Zhongshan mentioned by Liu Bei in the Three Kingdoms). As shown in the picture below
The reform regime that appeared before the establishment of the Eastern Han Dynasty was also separated from the same branch as Liu Xiu. Later Liu Xiu established the Eastern Han Dynasty and became Emperor Guangwu. As shown below
As for why we should inherit the country name of "Han". I think the reason is: the purpose of carrying forward the Han Dynasty and returning to the Han Dynasty. This is just to comply with the people's wishes, to comply with the will of the people, and to go with the current, so as to rule well!