Endless search for updates on 2014 MacBook running 11.7.10

Big Sur is the end of the line for that model Mac, but I agree it's not ideal to have Software Update apparently stall at "Checking for update" with no end in sight.

What you describe was a problem a while ago. Solutions known to be effective include restarting the Mac in "Safe Mode" and checking for updates while in that mode, and / or resetting NVRAM.

How to use safe mode on your Mac - Apple Support

Reset NVRAM on your Mac - Apple Support

Try both.

2. Do you see it as a security threat to my newer machines sharing the Apple ID and WiFi?

No.

NVRAM contains information retained across shutdowns or loss of power. Settings such as the Mac's startup disk, volume controls, that sort of thing. Reset NVRAM on your Mac - Apple Support explains some of them, but it is not an exhaustive list because "These settings vary by Mac and the devices you're using with it."

Possibly corrupt NVRAM contents have been associated with the endless "checking for updates" problem you experienced, which I experienced myself at the time. It may also be the reason it arose for you only recently. I no longer recall what it took to fix it but I am nearly certain I reset both SMC and NVRAM and it ceased to be a problem.

1. You check for operating system updates via System Preferences > Software Update, not with a web browser. The last update to MacOS v11.7.10 occurred in Sept 2023.
2. No. Provided you are using common sense passwords for your Apple ID, Wi-Fi SSID, and network devices.

On July 29, Apple provided customary point updates to Ventura, Sonoma, and Sequoia — all currently supported operating systems. Very likely there were security updates in those releases in addition to feature and bug fix updates.

FYI, while that Mac can't run anything later than Big Sur, and Big Sur isn't eligible to run the latest version of Safari, Big Sur is enough to run the current versions of many third-party Web browsers.

These include Firefox, Brave, Opera, Microsoft Edge for Mac, and Google Chrome. (The current version of Firefox will run on Catalina, and there is an ESR version of Firefox for Sierra – Mojave that got security updates up through this month. So it might be a few years until Mozilla decided to completely abandon Big Sur Firefox users.)

jack3030 wrote:

I reset SMC repeatedly (to ensure I was doing it correctly) and [only] after several tries the update check has completed. Thanks again.

You're welcome. Thanks for the update.

Determining an SMC Reset was in fact correctly performed can be difficult. Sometimes you will get two startup chimes, sometimes a MagSafe LED will change states, sometimes they don't, but the fact the updates finish now is an indication of success.