It wasn't just the Democrats that tried to prevent people on the "lists" from being able to buy firearms. Cornyn also introduced a bill preventing anyone on the list, or had been on the list within the past 5 years, from buying a firearm. His bill satisfied the NRA and the ACLU's concerns by requiring the government to prove to a judge within 72 hours why the person should be permanently barred from purchasing. Harry Reid didn't like that the bill was supported by the NRA and Joe Manchin didn't like the due-process requirement. It, too, failed on a party line vote.
There were 4 gun-control measures brought for a vote, 2 by the Democrats and 2 by the Republicans; all failed along party lines.
"Not all Democrats are for a complete ban on guns..." While this statement is true, there are enough Democrats in positions of leadership that want to ban some sets of guns that, taken in aggregate, represents pretty much a ban on the most useful and popular guns. "Australian-style gun control," popular with both Clinton and Obama, is a ban on semi-automatic rifles and shotguns. "Assault weapon" bans are a ban on the most popular rifle in the country.
By the way, there's significantly less regulation on operating a vehicle than in operating a firearm (treat guns like cars). There's also a lot more paperwork involved in buying a firearm than in getting a hunting license, at least for the states in which I hunt.