Can Lightroom Classic be used to re-organize files and folders that are already in the library?

"Can I quickly re-organize the big bucket folder..."

Yes, but "quickly"- no.

My take on the question: I would start by somthing fairly simple- Move photos into 'Year' based parent folders.

An Example- Find all photos dated in 2016  and move to a 2016 folder within {Photos} parent folder.

1) Filter All Photographs for Metadata 'Date'- Selecting the 'Year'

2) Select all the '2016' photos [Ctrl+A]

3) [Right-Click] on the parent {Photos} folder and [Create folder inside "Photos"]...

4) Check the option [Include selected photos]

5) Click [Create]  and all 2016 Photos are MOVED from the huge single folder to a '2016' folder.

6) Repeat the procedure to create and move all photos to 'Year' based folders (2017, 2018, etc)

7) As time permits you could then Move photos into 'Month' sub-folders by the same procedure- depending how OCD you are.

This method preserves everything done in the Catalog- Keywords, editing, Collections, etc. etc.

Basically, anything you can do in the Finder (make and name folders, files, move and delete them) you can and should do in LR as it then 'knows' what you've done. So basically the answer is yes. Easily, quickly? Not any more than doing so outside of LR but at last doing so in LR will instruct the database of what you've done. So you could make new folders for images, move them, rename them etc, all with LR. Quickly? Probably not. 

One large folder does have the drawback that operating system operations within that folder will be slow.

Re-organizing to date based folders (by using Lightroom Classic, not your operating system) would have to be done manually, one date/one folder at a time. Only you can decide if it is worth the effort.

Re-importing will cause you to lose certain work done previously, such as collection membership, virtual copies, photo books, slide shows, sequence of edits and a few other things that I can't remember. If you perform the Ctrl-S or Cmd-S method as described by @john beardsworth you will not lose your edits, keywords, captions etc. but the items I mentioned in the previous sentence will be lost.

Not really.

However, what you could do is remove the photos from the catalogue and re-import that big folder, choosing the correct option in the Import dialog - in this case Move and getting the Destination panel pointing to your date-based structure. Before doing this, select all the photos and do a CmdS (mac) or CtrlS (win) which saves most of your LR work back to the photos. Try this on a few photos first, and you'll see what you lose (stacks, pick flags, history steps, collections, virtual copies).

So that's really a yes, but not as straightforward as it might be.