What distributions have you tried?

MX Linux
Mint
Debian
Manjaro
Ubuntu
Pop!_OS
Zorin
Lite
antiX
elementary
PCLinuxOS
Kali
SparkyLinux
Puppy
Q4OS
And more...

Had to use some parameters...

32 bit
End of Life (Support)
And the killer... Working WiFi Driver

Using a Ventoy USB stick with about 25 .iso images on it to choose from.

It sounds like you've hit all the ones I would have suggested. Apparently this laptop doesn't support 64 bit?
Rather than frustrate yourself I would suggest getting a USB wifi card. I've had good luck with those in situations like yours. They aren't too expensive and can be found easily on Amazon. I have forgotten which brand/model I've used or I would point you to a specific one.
I did see one person on the Linux Mint forum say that they had to unplug their ethernet cable/card for this Broadcom wifi chip to work.
Its been years since I have had any issues in Linux with hardware. I guess I got lucky and never came across this chipset in my various systems over the years.

On my personal machine I dual boot with Windows 10 and Ubuntu Studio.

Ubuntu Studio is a dream. Easy install, updates, everything i need and want.

So... I assumed that all versions of Linux were like that. Rookie mistake.

Some installs reminded me of Space Invaders (DOS), and others would see the wifi adapter and connect me with my wifi network, but when I tried to put in my passkey, would grey out the connect button half way through and not allow me on.

Others got on, but would not allow me Internet access.

It's been quite eye opening.

You have to remember that the issues you're dealing with are not the fault of Linux. For instance Broadcom could have at any time taken the time and effort to release a working driver for Linux for this chipset. They apparently have not bothered. That leaves it up to the greater Linux community to try and figure how to make this chipset work. Without help and support from Broadcom thats a tough job.
In my experience what does work in Linux tends to work far better than the same gear in Windows. I've been a Linux user since around 1997 and no way would I go back to Windows. In addition Microsoft is far from being a company that supports progressive values Their treatment of other companies and Linux has been abysmal.
Currently I use PCLinuxOS and MXLinux. Both work really well. PCLinuxOS is a "rolling release" version of Linux. I've been using the same system for several years now with it updating itself all along the way.
I've used the usb wifi adapters in the same situation as yours and have had good luck with those. I believe the brand I used was "Edimax". Not sure what chipset but a search for "Linux USB wifi adapter" on Amazon should turn up something quickly.
With Microsoft prematurely obsoleting systems before their time there are plenty of dirt cheap computers/laptops out there to choose from. Even with this laptop you're having trouble doesn't work out its not hard to find another more suited to Linux.
Shopgoodwill has tons of them. I haven't bought a new laptop since the 90's. Keep finding free or next to free ones as I go. They all have worked fine with Linux. Good luck on the project.

I may bitch a lot, but I'm secretly enjoying this frustrating treasure hunt.

In my lifetime, there have been 2 "Evil Empires"... Starbucks (don't know why) and Microsoft.

And a year or so back, I was asked by an elderly lady if I could get her laptop to work again.

She was a Windows XP user, and it had to be wiped out and redone. After saving what personal files she had, of course.

I told her about how XP was going away, and her OS and browser would slow down and eventually stop working... she would no longer get security updates, patches, etc...

She didn't care. She told me she was not going to lay out the money for a new PC because she wasn't going to be around much longer.

I told her about Linux, and she said she didn't want to learn a new OS for the same reason.

She made sense, and I thought about how many other elderly folk like that are in the same situation.

And the bricks and roadblocks I encountered just trying to get her laptop back to working condition and as up to date as possible were mind-blowing.

My hatred of Microsoft grew with every keystroke. Took several days and a number of long nights.

I don't blame Linux at all. In fact I stand in admiration of all the hard work that goes in to he various versions and flavors.

And if that pisses off Microsoft, all that much better.

I just keep thinking, "I must be missing something, something really simple and small."

And... it's just kinda driving me nuts.

But don't worry... it's a short drive.

One, an old NetGear. And another a new TP Link

Some distros saw them, some did not.

NONE, let me connect to my wifi network or the internet for more that 30 or 40 seconds.

Not sure what to do with it.

Not the one I'm working on.

That one has no internet.