Unfortunately, I am old enough to agree with you that all late night talk show 'hosts' did skewer politicians.
The HUGE distinction is that they were mostly agnostic about which party got roasted (your own example of Conan) and weren't nearly as vitriolic as they are today. There's funny and there's mean-spirited. A bully who thinks he's being funny (can you say Stephen Colbert? Or Jimmy Kimmel?) sooner or later gets their comeuppance.
Today's 'comedian/hosts' target only one ideology...the one they disagree with. 46 was a motherload of material for humor if they'd only let go of their TDS.