They do.
American Trends Panel: Pew Research Center’s online probability survey panel, which consists of more than 12,000 adults who take two to three surveys each
www.pewresearch.org
The most consistent voting demographic in the US skews Republican too.
Politicians go where the votes are. That's what makes political boycott counter-productive.
I find the fixation on the Democrat-Liz Cheney link-up somewhat silly, considering that this is only happening in the context of discussing the state of Democracy in the US, something I would expect every sane person, of any political conviction, to be worried about. Nobody should be comfortable with a democratic system in which a Donald Trump presidency is a reality, and the voting base punishing democrats because they highlight that those they disagree with share the same worry is borderline sensationalistic.
The irony is, Democrats being repudiated by their traditional base for being seen with Liz is exactly what may make them move towards the right, especially after seeing the left's rejection of the Biden administration after he started implementing many of the economic policies they championed (better wages, investments in blue collar sectors, labor protections, college loan cancellations, etc...).
That's 3 out of 4 nonvoters didn't have college degrees between 2018-2022 (this number isn't going down soon); if Democrats continue to be painted as the party of the "elite," despite their very populist policies, it will be much harder to convince these nonvoters to support them in future elections.
I think folks who support Democratic policies must take into account how much their criticism of Democrat politicians feeds into the negative perception the less politically engaged public has of the party, and how much that can hurt their electoral prospects. At the end of the day,
what we want is someone in congress to pass the bills you like, and you want someone in the White House to sign those bills into laws.