One criticism Liu Bei sometimes gets is if he was loyal to the Han, why did he take the throne? Now there are several answers used on that (he deserved it, practically, Han had done this before, yada yada) but I wonder if part of it is that the Han emperor in themselves were no longer important, that changing who ruled was a fairly minor issue by 3kingdoms, that the Han was far far bigger then Son of Heaven even for loyalists.
Liang Ji murdered Han emperors and the reaction of Han loyalists seems to be quite small to his regicide, they didn't like Liang Ji for a lot of reasons but off the top of my head, the regicide didn't really anger them. He Jin's faction overruled Ling's wishes for successor, Dong felt able to go "ok Han emperor needs to change", the Coalition seriously considered replacing Xian with one of their own, Xun Yu was content with Cao Cao's sometimes rough treatment of Xian until a signal that Cao Cao might want throne himself.
Clearly it mattered to some Han loyalists at way Cao Cao treated Xian, the coalition idea was blocked, Liu Bei faced objections so I'm not saying it was universal or that the Son of Heaven was unimportant to them but was it less important then we tend to think? That if push came to shove, replacing one with another was not a big a deal as it might seem?