What those fellows did in Georgia wasn't a lynching at all, so it wouldn't be covered by this new law.
Just giving you the concept of how a Federal law can be used to enhance the punishment given by a state court on a separate charge. It can be more beneficial as Federal charge, not charged by the state, especially in times like these and people with views, once simply abhorrent vying for power and influence as they are.
Some groups, ones that would support committing hate crimes are welcome now in the Republican Party, with a past president tiptoeing around, avoiding condemnation. If not welcome, at least tolerated to the point, big name republicans actually show up to court their vote.
Lots of changes of what they will tolerate, now that conservatism is not the defining characteristic. In the 60s, 70, 80s, 90s and up to 2008, you would never hear a Republic speak out complimenting the leader of the USSR or Russia, while Russia attacked a free country. It just didn't happen. Now not only do you have them speaking out for Putin, but big name right wing news/infotainment/talk show host(s), as well, to the point they are must broadcast as part of the state ordered Russian propaganda broadcast to the Russian people.
Back to the hate crime statutes at Federal level. Their use pretty well guarantees a hate crime criminal will not be pardoned and released of state crimes, simply because a Governor my might come to office, sympathetic to a hate crime supporting group, supported his campaign. I have no doubt, it is thought like these that got the measure passed by such a wide margin in both houses of Congress.