There's only one solution to time travel paradoxes that resolves all of them, makes some kind of sense, and doesn't require generating an infinite number of additional versions of the universe - single coherent timeline. The universe as it is now is the sum total result of all time travel that will ever have happened to a point in time before the present. You can't go back in time and kill Hitler because you didn't, which means any attempt you might have made to do so clearly failed. Likewise for the grandfather paradox, if you go back in time to kill your grandfather, you already failed in your mission.

Awful views aside, one of the things I gave JKR credit for was doing a decent job of adopting this model of time travel in Prisoner of Azkaban. Then she threw that out the window for Cursed Child because she's terrible at any kind of world building consistency.

I know how history works. I don't need a time machine to know that a country in economic dire straits will find someone to blame. I just need to look at contemporary US politics. If you're ever in doubt start a war, preferably somewhere in poverty so you don't get your asses handed to you, inevitably miscalculate, and get said ass handed to you. Wait 20 years and then do the same thing all over again.