Key takeaway
Real time savings come from removing specific tasks the old process required, not from vaguely "being more organized."
"Will this actually save time" is a fair question to be skeptical about, because plenty of tools do not save time, they just relocate the same manual work into a new interface with better colors. A heavy tool that needs constant configuring, manual status updates copied between screens, and its own maintenance routine is not a time saver on net, whatever it claims on its homepage.
Where the time actually comes back
Real time savings come from specific, nameable things the tool removes, not from a general promise of being more organized. A first draft of scope and milestones from a short description or an uploaded document, instead of an hour staring at a blank page. An automatic reminder email to whoever owns an overdue task, instead of a PM spending Friday afternoon manually chasing five people. One place to check a project's status, instead of reconciling a task board, a document and a chat thread by hand before answering a simple question. Each of those is an hour or more, every week, that used to be spent on manual translation rather than actual decisions.
The honest caveat
None of that time comes back automatically. A tool only saves time if it is actually kept current, the same discipline a spreadsheet needed, just with far less friction attached to it. The realistic promise of a good tool is not that project management stops requiring effort, it's that the effort goes toward the parts of the job that actually need a person's judgment, deciding what to do about a risk, not the parts that were always just manual bookkeeping wearing a project management label.