Step #0 to any significant policy reform is eliminating the Senate filibuster, followed by step #1: controlling both houses
and the presidency.
The only truly radical policy that has come out of Congress in decades was Obamacare, and that was precisely because the Democrats were just barely able to assemble 60 votes in the Senate. (Most Democrats wanted single-payer, or at least a single-payer option, but a bill including single-payer couldn't clear a 60 vote hurdle as at least one Democrat, Joe Lieberman, publicly opposed single-payer.) And even then, with the death of Ted Kennedy they immediately lost the ability to beat a filibuster, which is why the House was forced to pass the Senate bill verbatim, which was unusual. Usually there's a reconciliation process for bills passed between the two houses, after which each house has a pro forma vote on the reconciled bill. But even a pro forma vote on a reconciled bill is subject to a Senate filibuster, with the exception of purely budgetary bills, which Obamacare clearly wasn't. (There were subsequent changes passed as budgetary amendments, though, both by Democrats and later by Republicans--e.g. elimination of the individual mandate penalty.)
And don't forget, precisely because the Democrats passed Obamacare, the Republicans swept the 2010 election, taking the House and taking 6 Senate seats from the Democrats--who went from 56 in 2010 to 53 in 2011, not counting the 2 seats lost earlier in 2010--in December 2009 Democrats had 58 votes plus 2 Independents. The 2010 election was the biggest rout in 60 years.
Unless and until the Senate filibuster goes away, you can't really make any sweeping claims about what policies a party could or would actually enact. Because of the filibuster the parties are beholden to a handful of members. I'm not entirely sure the filibuster should be eliminated, but if I were a Senator I probably wouldn't stand in the way, either.
If you follow the politics, both houses and both parties could easily enact significant immigration reform on a simple majority vote. Albeit not radical reform so long as the filibuster existed. However, immigration has proven a tremendous wedge issue for Republicans, which Trump leveraged to great effect. Moreover, even though a vast majority of Americans and Congressmen wish to change the status quo, any specific, concrete policies lead to a lot of heated disagreement because it turns out different people have conflicting motivations for wanting reform, especially within the Republican camp. Moreover, the Republicans learned a lesson watching Obamacare--if you can't enact perfect reform (i.e. Democrats being forced to compromise on single-payer), your own supporters will crucify and disown you. A big part of the Republican 2010 sweep was upset Democrats walking away because they didn't get their preferred option. So especially for Republicans, immigration reform is a lose-lose-lose.