Have you tried fixing the current Windows instead of reinstalling it? I'd start by running "DISM /Online /Cleanup-Image /RestoreHealth" (without quotes) in admin privileged Command prompt or Power Shell and after that "sfc /scannow" to see if they fix anything.
If that proves unsuccessful, upgrading to your current (same) version of Windows from a memory stick created by the instructions on https://www.microsoft.com/en-us/software-download/windows10 should leave your programs and files untouched.
Update:
After working w\ Hitech yesterday I got around to checking into the error 87 in DISM.......invalid parameter. I went back over how I typed it in Command prompt and found I had fat-fingered it.....typed it in correctly & ran it........all ran successfully then again ran sfc /scannow.....and again all came back clean but when I went to test the game again I got the same issues so I finally broke down last night & went thru the process. I went ahead & updated my graphics card drivers to the latest vers then downloaded the Windows Media Creation tool, set it up to reinstall Win 10 Home x64 keeping my external files & programs intact & reinstalled the OS. All went well w\ no hiccups or issues so she came back up clean w\ all still intact & functioning well. Was after 2:08 hrs MDT so I shut it down & went to bed.
Got up this morning & tested AHIII to see if we got anywhere (had manually updated game yesterday).....game would start up fine, went online to the arena clipboard fine, was pinging the AHIII arena server just fine but when I clicked to go into any arena my box wouldn't allow the AHIII server to download the terrains to my box & allow access to the arenas....would default back to the AHIII menu clipboard. So I got an idea to move the entire game folder to another folder on my E:\ drive so I could then manually reinstall AHIII back to the same syntax so it would be a fresh install (not written over) but to my box it would look the same. Got that done, made all the geek tweaks to the Settings folder then ran it. Game started up just fine, went online to the arena clipboard just fine, pinging the AHIII arena server just fine. Tried to enter the Open Melee arena.....it downloaded the terrain/connected up just fine....all looks well....but then backed out & tried to access the rest of the arenas (which reside on the same server so pinging the same IP address) to test.....my box blocked all the other arenas but will now allow the Open Melee arena to download\run just fine.
Hitech informed me that he was pushing down a game patch this morning so that we could test to see if the patching interference was resolved\still present. Started & signed into the game after some time, saw the dialog box indicating that a patch was available, clicked it to run it & my box promptly blocked it as seen in the video I posted so this issue still exists. Hitech had also sent me the link to get it manually if it did get blocked so I manually patched up to patch 14 then went to start the game.....game started up just fine, went online as stated earlier just fine, accessed the Open Melee arena just fine, ran just fine but still would not access any other arenas except the Open Melee arena. Just to clarify, all this is happening w\ both Windows Defender AND WebRoot Anywhere AV's
fully active w\ AHIII setup in both to allow the game full access thru the firewalls (checked & verified MANY times) & it appears that both firewalls are allowing AHIII clear passage so this is the quandry I have going on. The reinstall of Win 10 doesn't appear to have had an effect so it would "appear" to be software related....something on my box is blocking HTTP port 80 when AHIII servers are trying to access it to run the patch updater or when AHIII arena server is trying to download terrains\connect the arenas to the game client on my box but I have absolutely no idea of what it could be as I haven't installed any new software since I upgraded the CPU\mobo back in Dec '19....why I suspect some Windows update as a cause but I have no idea as to which 1.....the reinstall won't rectify this either as it will reinstall all of them as well to the fresh install.
So this is the monster I'm trying to chase down within my box but now at least I can play the game again!

Gonna take awhile but eventually will be found.
Could be something in his router settings possibly blocking port 80
Depending on if he is using certain router features..... Could also happen in Windows10's firewall/ network settings if he has accessed it and possibly accidentally changed settings
Just some thoughts
Best of luck with it
Been going down that path as well TC. I have checked, rechecked, checked some more, reset my modem\router several times....router setup is default & has been working fine (outside of Centurylink recently having issues at their DSLAM & our router getting too hot due to the wife crowding too many pictures around it blocking airflow to it--uses passive cooling--cleared some space around it then reversed the ceiling fan rotational direction to output airflow across the ceiling & down the walls to create air movement around the modem\router.....fixed the overheating) so can't find any issues there. Now as for my onboard LAN's network settings I'll have a look at them but I wouldn't know which setting(s) would try to block access to a specific port for any application much less just AHII as the only settings I changed in there are the transmit\receive bit widths. I'll go back & reset all this back to default to see if this changes anything but I kinda doubt it. I did run a traceroute to the AHIII server IP address from my Centurylink router & it failed a couple of times but finally went thru & completed....but running Ping Plotter from my box to the same address at the same time completed every time so I just might look into trying another modem\router if I can't find anything in Windows but all hops that were showing failing were not located in my general area of the state (including the Albuquerque area) so I can't see the router being bad at this time anyway.
I did recently update the LAN drivers to try to see if updating the drivers would fix this but it did not.
This has been for me a tough nut to crack w\ as of this time going thru a clean wipe\reinstall of OS then manually updating it along w\ individually installing every piece of software to try to catch the culprit is not something I'm looking forward to be doing just yet so I'll just keep nipping at it......at least Hitech helped to confirm what the issue consists of so this narrows it down some.
Now yesterday working w\ Hitech we got Win Defender to shut down & thought we had WebRoot shut down as well to get them out of the way while running tests but when I rebooted my box sometime later Win Defender still showed it was shut down but WebRoot restarted itself somehow, even though I have it checked in the WebRoot Advanced Settings to allow the AV to be manually shut down......
May go in & wipe WebRoot to see if anything changes.................
Well any way now y'all know..................

PS--Already informed Hitech but I ain't got no VPN going in my system....neither hardware or software....FYI.