Is There a Free Way to Fix Google Takeout Dates?
Yes, several. Here is an honest look at the free options and where the paid one earns its money.
Yes, genuinely. exiftool is free and can do all of it, if you are willing to write the command and handle Google's awkward sidecar names. GooglePhotosTakeoutHelper is free, open source and well maintained, and it is good at reorganising an export, though it sets file dates and points you at exiftool for the step that writes the metadata inside your photos.
If either of those suits you, use them. They are good software and this page is not going to pretend otherwise.
This app is free for your first 500 photos, which is enough to fix a small export outright or to test a large one properly. Past that it is $9.99 once, and what you are paying for is not doing the job at all, but doing it without a terminal, on a copy that leaves your original untouched, with a list at the end of every file it could not match. If that is worth ten dollars to you, it is there. If not, exiftool is genuinely free.