I don’t know why there is any debate on this subject. Time travel exists. Anyone who reads or even daydreams experiences it several times a day.
We go places. I have been to the middle ages. I’ve experienced foreign lands in other times. I’ve met people long gone. I have my jet pack, and have used it more than once, believe you me.
I have seen Mount Vesuvius erupt. I have been at ground zero for the first atomic bomb and for the last. I’ve witnessed the birth of Christ and the deaths of kings. I remember when dodo birds weren’t extinct. I have visited the first colony on mars.
The thing people seem to be so upset about regarding time travel is that they don’t get to bring their bodies along, and they don’t get to arrogantly meddle in time lines.
To that I say, tourists can always find something to whine about. Get over it. Sit back and enjoy the ride.