I keep checking for software update but no update October

Most likely because you have a budget phone from Moto. Moto is notoriously bad at sending out timely updates even for their flagships, so it will be much worse for their budget phones (which will probably be lucky to get even 2 or 3 security updates, much less a major Android version upgrade.

JR Raphael (my favorite Android/tech journalist) does an annual report card grading how each major phone manufacturer does with updates. Moto is always at the bottom. https://www.computerworld.com/article/3606708/android-11-upgrade-report-card.

The 2nd screenshot you shared is from the main Android Source Code site, which is the pure Android code. Google releases it publicly, and Pixel phones always get it pretty much as soon as it's released. But it's up to the other phone manufacturers to test the code and tweak it to make sure it runs correctly on their phones. A lot of them don't have the resources nor the desire to put in that work.
 
I did apply to automatically OS upgrades still my smartphone didn't automatically and download OS upgrades no avail, how do i fix it?
Screenshot_20211102-081000.jpg
 
Perhaps I'm not making it clear: I don't think your phone has any official updates to install, and so that's why nothing is installing. That screen you're showing is just an option to select when to install an update ONLY IF one becomes available. But because Moto is so bad at releasing updates, especially for its budget phones, I really don't think you'll see any.
 

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