I’ve decided that the only way time travel to the past can possibly exist, whether in fiction or reality, is that when someone or something is sent into the past, an alternate timeline immediately comes into existence.
Let’s say some scientists are working on a time travel device. To test it, they decide to send a blue widget 5 minutes into the past. With no alternate timeline, they would see a blue widget suddenly appear 5 minutes before they start the test. Success! But now they have a dilemma similar to the grandfather paradox. Either they go through with the test 5 minutes later, and create a closed loop in which the scientists at the T-5 mark would actually have gotten infinite blue widgets, or they don’t perform the test, and create a time paradox.
As opposed to an alternate timeline scenario. At T-5, the scientists see a blue widget appear. They know the machine works. They can choose to go through with the test, in which case they create another alternate timeline in which the scientists got two blue widgets, or choose to not perform the test, knowing that it works and they created the alternate timeline they are now part of.