That's correct. Get-out-the-vote efforts are seen as partisan. I volunteered to help register people to vote (as a non-partisan without any messaging at all) and was verbally abused by Republicans on the street.
Republicans also regularly impose laws to disenfranchise voters. An example is Georgia's law where voters are purged from the rolls every 3 years. Minority voters often only vote every 4 years.
The list goes on and on. These voter suppression tactics are very, very targeted at minorities and Democrats, often to the point that the Supreme Court has to strike them down as unlawful.
Yeah, basically if everyone always voted, the Democrats would generally win. That's why Republicans don't want to encourage voter turnout and also want to suppress ease of voting which usually harms poorer people who tend to be Democratic.
Perhaps, but it'd then just be on the same level as any other broadcast medium that puts out such a message. I presume the more partisan of the cable news channels are still allowed to say "hey everyone, vote!"?