What's the error message that's being displayed when the download fails?
I'd guess it's probably something about the version of OS X being too old?
According to Google, OS X 10.9 is required for the current version of Chrome, which means you'll need to select and use a different web browser such as an older version of Firefox, or find an older version of Chrome that supports 10.7.5, or a different operating system, or (if your Mac supports it) upgrading to a more recent version of macOS such as 10.12 Sierra. Or replacing the Mac with a newer model, though that's not cheap.
Testing and maintaining software for older versions adds costs and complexity (also with some alternatives to Google Chrome that might still work on 10.7.5), and older versions of OS X will also generally have problems connecting securely to remote servers, too.
Unfortunately, some of the downloads from sites other than at Google that might claim to be Chrome could well contain malware. I don't immediately see a Google download site with older versions, either.