Exactly for this purpose we have developed a new online service
We had the same problem. Our clients are sending us very large files quite often. The DriveUploader service supports the transfer of files with up to hundreds of gigabytes!!! (yes, we have tested this and received such a large data from our customers).
The service comes with a free plan - where you create a time-limited private upload link, which is sent by email to the persons who upload any large data to a folder of your choice in your Google Drive. They don't need to log-in anywhere, just use drag&drop into their web browser.
You can also embed the service in your websites via iframe and brand it with your company logo, with minimal technical knowledge.
The uploading component comes with a nice look&feel:
I hope you (and others) find this service useful...