If you're using a stock android device, the OS on your phone still has permissions to read and write to storage, by necessity. If what you're concerned about is privacy, you have very limited ability to set storage scopes if you don't trust the OS, and this doesn't really change if you install an app.
If you're using fossify file manager or any other file manager, you've given that app+the default Files app access to your storage. This is not more private. Most of those similar apps are essentially just skins on top of the default manager (which I suppose could be useful). This only really adds attack surface and doesn't have any meaningful privacy benefits, and potentially some detractors depending on the app you use.
If you don't trust the operating system and its utilities, the best option is to find an operating system you trust, and not to just install new skins on top of existing apps.