Is there an app or script to Move Files in Finder with a mouse click?

On the Move Action, click the Options button, then check the "show this action when the workflow runs."

That will allow you to pick the folder each time. The one set in the workflow will be the default.

Instead of an application, set it as a Quick Action and you can ctrl-click on the files to move.

You don't need the Get Selected Finder Items. The workflow gets them automatically.

1) No. If you only have 1 or very few, I don't think there is a menu. But, that may have been from the early stages of the Quick Action menu and now they are all inside.

2) No. It's not an open/save panel. It's a widget created for the action.

3) Yes. They show up in Keyboard System Settings > Keyboard Shortcuts > Services (they were previously lumped in with Services, but are now called Quick Actions).

There are several categories in that section. This one would be under Files and Folders

This may be of interest, also: Create keyboard shortcuts for apps on Mac - Apple Support

I think I need to learn Automator!

I would try to learn Shortcuts, instead. Since Shortcuts does essentially the same thing and is "shiny," it might replace Automator at some point. I would only resort back to Automator if it can't be done in Shortcuts.

An alternative. Get XMenu which is from a reputable developer and is free on the App Store. You can make shortcuts to popular folders and also display them in a hierarchical menu (e.g. HOME starts at your main drive level). Open the other folder and drag the files from one to the other.

I also found this which sounds like it does what you want: https://www.makeuseof.com/automatically-move-files-from-one-folder-to-another-windows-mac/

Scratch was just the name of a newly created directory I made to create this example app. You can specify any folder you like.

I put the app on the dock so I could launch it with a single click.

I then used the Finder to select files I wanted to move, then clicked on the move app in the dock and all the selected files were moved to the specified directory (in my case that selected directory was called "scratch"; you can specify any destination directory you prefer when you create your Automator app)

Applications -> Automator

can create an app that does what you want.

You can move files by selecting them, press command+c, navigate to the location you want to move them to and press command+option+v.

You didn't do what I suggested. When you press Command Option V, the file(s) are copied to the new location and deleted from the previous location automatically.

Just to be clear, Command V would create a new copy and leave the old file. Command Option V creates a new copy then deletes the old file.