No local host connection = no boinc.exe running (at least as far as I can tell).

Yup, that's a good takeaway. Conversely, boinc.exe has to be running in order for connection to be made.

10-Dec-2019 04:47:11 [---] Starting BOINC client version 7.14.2 for windows_x86_64
10-Dec-2019 04:47:16 Initialization completed
10-Dec-2019 04:47:16 [---] Suspending GPU computation - computer is in use
...
10-Dec-2019 04:49:23 [---] cc_config.xml not found - using defaults
10-Dec-2019 04:49:23 [---] Starting BOINC client version 7.14.2 for windows_x86_64

So boinc has run, for just over two minutes. That behaviour can happen if the cooling system is completely broken (enough time to reach shutdown temperature), but it's unlikely in this case - (a) overheating usually crashes the whole computer, not just one app, and (b) with no science apps, there's not really much heat generated - not even benchmarks.
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Note this is a fan-less mini (like an Intel NUC) Asus PN40 to be precise. I checked and it's been out for less than 2 years (18 months or so, I think)
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@other helpers - didn't we have a problem with one of those low-power Intel chips failing to run 64-bit SSL routines? Might be worth a try with a 32-bit BOINC while we look up 'Celeron(R) N4000 CPU'.

Open BOINC Manager (it doesn't matter in this if the client runs or not), View, Advanced view, Options, Other options, General, if it isn't checked then check "Run the client?", Save. File->Exit->acknowledge to exit. Next restart BOINC Manager, does that start the client and keep it running? ---No joy, already set.---
Else open Windows Event Viewer (Start->Run, type eventvwr.exe and tap Enter, then wait for it to populate), in the Windows Logs, Application check through the Application Errors if there's any mention of what happens to boinc.exe, if you find something post that here.

----many instancs of the following:

Faulting application name: boinc.exe, version: 7.14.2.0, time stamp: 0x5bbe9f91
Faulting module name: LIBEAY32.dll, version: 1.0.2.7, time stamp: 0x56d5fc8e
Exception code: 0xc0000005
Fault offset: 0x00000000000e6543
Faulting process id: 0x9e0
Faulting application start time: 0x01d5af57f0b02b28
Faulting application path: C:\Program Files\Boinc\boinc.exe
Faulting module path: C:\Program Files\Boinc\LIBEAY32.dll
Report Id: be3de9d0-d9bb-4e68-8668-e5e33737dee4
Faulting package full name:
Faulting package-relative application ID:
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In BOINC Data directory, stdoutdae.txt, anything of note in there?
Same directory, in stdoutdae.txt, any mention there what happens?
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I included the stdoutae.txt lines of last attempt in last post. I note this:
10-Dec-2019 04:49:23 [---] cc_config.xml not found - using defaults
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How did you install BOINC? ---from the boinc site at Berkeley.--- With your user account ---many times---, or did you use "Run as administrator"---several times after regular (administrative user,me) failed--- on the installer? The latter is not advised, as this will cause all kinds of permission problems. An administrator account is needed for the installation of BOINC, but not the super user administrator. And did you check "Allow all users on this computer to control BOINC" in the installer? --- always---

Do remember that BOINC consists of two components, the Client and the Manager. They communicate over TCP/IP, port 31416.

First check - is the client running? Dies boinc.exe appear in Windows task manager?
Second check - can that purely internal network traffic pass through your security device/software?
Third check - can boinc.exe (only) communicate externally over both port 80 and port 443 (http and https respectively)?