Another Cursed Spinning (Network) Wheel…

Try first to optimize the range of the mobile network, see this Apple article:

If you see No Service or Searching on your iPhone or iPad

If you see No Service or Searching on your iPhone or iPad - Apple Support

Check also the Wi-Fi network, see the following Apple articles:

Connect to Wi-Fi on your iPhone, iPad, or iPod touch

Connect to Wi-Fi on your iPhone, iPad, or iPod touch - Apple Support

If your iPhone, iPad, or iPod touch won’t connect to a Wi-Fi network

If your iPhone, iPad, or iPod touch won't connect to a Wi-Fi network - Apple Support

a wifi outside my home or office (ie owned by a “3rd party”).

That means you are using Public Wi-Fi at that moment, see this Apple article: Use captive Wi-Fi networks on your iPhone, iPad or iPod touch

Use captive Wi-Fi networks on your iPhone, iPad or iPod touch – Apple Support (UK)

Jacobi13 wrote: "I wasn’t dodging, I mentioned “during the recall”. To be specific, it was in the last month the free/discounted battery replacement was offered (I didn’t hear about the program until late).

So, Dec 2018…

[I just want this clunker to last until the 14 is released…]"

I can understand wanting that clunker to last. If the battery had been replaced more recently, say in Dec 2019 or 2020, I would have suspected that Apple's maximum capacity algorithm was inaccurate. (Mine has been stuck on 86% for a year!) But a 3½ year-old battery is consistent with your 79% capacity — and Apple considers a reading below 80% indicative of a weak battery.

...You can know for sure if that's causing the problem by replacing the battery — and you'll decide whether that's worth it or not. Anyway, below are the customary links related to battery replacement. Good luck!

iPhone Battery & Power Repair

About genuine iPhone batteries

Repair Provider Verification

(Articles found by searching Apple's Knowledge Base.)

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Jacobi13 wrote: "Can you elaborate? This is the first reference I’ve ever seen tying battery health (below a certain level) to a persistent effort to connect to a network only on the Home Screen and never when an App is open."

There's a first time for everything. We see a variety of weird symptoms on older model iPhones with old batteries. Nobody here can elaborate on how certain symptoms may or may not be caused by a weak battery. But having such a battery replaced can solve a variety of problems.

Suspecting a weak battery from age & use, we ask what the maximum capacity is and the answer is often <80% — like yours. Then we ask whether the battery is an authentic Apple battery and also the question I asked you: "...When was the battery replaced?" I await your answer.

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• wifi - mine, 3rd party, hot spot etc…

3 rd party what?

Disable Hotspot & VPN if the are turn On.

Jacob ~ Does your iPhone 7 Plus still have its original battery?

And what's the Maximum Capacity % here?...

Settings > Battery > Battery Health

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Jacobi ~ A “Maximum Capacity” of 79% isn't healthy and may explain the "Spinning (Network) Wheel". ...When was the battery replaced?

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