Online file tools upload your document, process it on a server and send it back. That model made sense when browsers could not parse a spreadsheet or a PDF. WebAssembly changed that: the same parsing and rendering code can now run on your own device at close to native speed, which removes the reason to transmit the file at all.
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For a file you would be comfortable emailing to a stranger, generally yes. For a contract, a bank statement or anything with personal data, the upload itself is the risk, regardless of the provider's retention policy.
Turn off your network and try the operation. A tool that processes locally will keep working; an uploader will fail immediately. That test is more reliable than any wording on the page.
For everyday files it is usually faster, because there is no upload and no download. Very large files are the exception — your device's memory becomes the ceiling where a server's would not.