Photo Mechanic isn't specifically making a request to download the files, however, it is accessing them no differently than it would any other folder on your computer. When you open a contact sheet, Photo Mechanic opens the files so it can parse the metadata. In response to this, the OneDrive process that runs on your computer kicks off a download, and now your folder is essentially acting like a very slow storage device.
I can also open any one of them up through File explorer and in about 3 seconds I am viewing it in PShop.
Photoshop opening the file is causing it to be downloaded as well. It's just that Photoshop is working with one file and not 4000. When you try to open multiple files in Photoshop, it also hangs until the download is finished (at least it did for me when I just tried).
There's no setting to change, Photo Mechanic is just not really designed around cloud storage services. I've heard of many users using Onedrive successfully to sync metadata changes between computers, but it's not going to do well if the files aren't on your computer.