When running a DOSbox GoG game on Mac, this is the sort of thing that I ought to be able to see: menu bar with emulation options, common keyboard commands, dialog prompts for things like quitting.

This works on games where I had installed it a long time ago, or copied it from my old Macbook. But when installing a new copy from GoG Galaxy or from the GoG website, the GoG wrapper is completely gone. Menu bar shows application as 'dosbox', no emulation options, no universal keyboard commands, no nothing other than the game... which can be quite an issue if I want it windowed or something.

I tried contacting support but I find it difficult to use the form for an issue that concerns the GUI wrapper rather than the game.

I haven't a clue what you're talking about unfortunately. DOSBox games on Windows have never had any of the things you've mentioned.

Mac OS Galaxy version 1 (not 2 closed/open beta), previous DOS game Wacky Wheels have overlay but if reinstalled today it'll not. You are not using Galaxy 2 at the moment. Is that correct?

Also I couldn't find Wacky Wheels in list of overlay supported games:
https://www.gog.com/games?feature=overlay&sort=bestselling&page=1

Lastly you may (or may not) find it useful to post in official Galaxy discussion thread:
https://www.gog.com/forum/general/gog_galaxy_official_discussion_thread/page1 see attached

So if this GUI wrapper is an aspect of the boxer, is it possible to use existing working boxer files and copy them across to a game where it's not working?

So as far as I can tell, what's happened is that GOG used to package their DOS games for Mac with Boxer for some reason. More recently they've started just using DOSBox directly in the way they do on Windows. No idea why they changed, but I have no idea why they bothered with Boxer in the first place. I think Boxer is a Mac GUI for DOSBox. You should just be able to install it and import whichever GOG games you want.
Personally I don't see the point, DOXBox isn't hard to configure and once I've got the setting I want for a given game I appreciate the directness of not having any menus or anything.

No that's not how it works. It's not a separate app, it's all contained within the game application which is downloaded.

I know that GOG used to package it in the installer. What I'm saying is that you can install it yourself: http://boxerapp.com/

I have no idea about Boxer shell for DOSBox, apparently some DOS games come prepackaged with Boxer-DOSBOX for which Galaxy have these GUI features. From your words recent game installs don't have the Boxer, therefore they might not have it. If you haven't tried copying the Boxer files from working game to non-working then try it now.
In any case it's GOG you need to ask about Boxer inclusion in recent version of Galaxy / offline installers for DOS games.

GOG's DOS games for Mac used to be run on Boxer. However, boxer has 32-bit elements and is no longer compatible with macOS 10.15 Catalina and will not run on that OS.

Therefore, GOG is updating their DOS games to run on DOSBox instead, which is Catalina-compatible.

The difference you note is the difference between Boxer and DOSBox.

If you are NOT using Catalina, you can download Boxer yourself separately (http://boxerapp.com/) and run the games with that instead. You will need to locate the game files within each game's GOG app to add them to Boxer. If you have any trouble with this, let me know and I can help you (it can be hard to find the first time you try, but once you know where to look you should be able to find them for all games).

Ahh, this makes sense! I have no wish to go to Catalina so I will try this out. Thank you!

It seems that there is a unofficial version of boxer compiled as 64 bit binary: https://github.com/alunbestor/Boxer/issues/76

Caveat: Can't test it because I don't have a Mac.