"Many Republicans seem most eager for steps that would make voting harder for groups that usually favor Democrats."
Like the dead, illegal immigrants and people who vote more than once, especially in multiple states?

The Dems also push hard for voting rights for convicted felons (even those still in prison), which leads one to believe those folks would be more likely to vote for Dems.
"I assume the Democrats would like to do the same in reverse, but they're not the ones currently pushing for new laws."
They have in the past, especially at the state level. They passed motor voter laws, then pushed for driver's licenses for illegal aliens. The current push is for a national law that would override the state laws they don't like, even though the Constitution leaves voting laws to the authority of the states.