I've done all the above to increase my storage capacity but still unable to receive the Software Update; help!

1.

Check iPhone type & iOS version -> Go to: Settings -> General -> About.

If it is an iPhone 6 Plus -> iOS 12.5.5 is the last security updates for iPhone 6 Plus, -> released 23 september 2021.

Then you can update to iOS 12.5.5

If it is an iPhone 6S Plus you can update the iPhone to iOS 15, see this Apple article:

iPhone models compatible with iOS 15

iPhone models compatible with iOS 15.4 – Apple Support (UK)

Go to: Settings -> General -> Software Update.

2.

You wil need about 7 GB device storage.

Try to free up iPhone device storage see this Apple articles below:

If you need more space for an update on your iPhone, iPad, or iPod touch

If you need more space for an update on your iPhone, iPad, or iPod touch - Apple Support

How to check the storage on your iPhone, iPad and iPod touch

How to check the storage on your iPhone, iPad and iPod touch – Apple Support (UK)

Clear Other storage on your iPhone or iPad

Clear Other storage on your iPhone or iPad - Apple Support

What's the difference between device storage and iCloud storage?

What's the difference between device storage and iCloud storage? - Apple Support

3.

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

Recommended settings for Wi-Fi routers and access points

Recommended settings for Wi-Fi routers and access points - Apple Support

If you need help with your Wi‑Fi password

If you need help with your Wi‑Fi password - Apple Support

4.

See this Apple article: If your iPhone, iPad or iPod touch won’t update

If your iPhone, iPad, or iPod touch won't update - Apple Support

You can do the update via PC, see this Apple article: Update software on iPhone, iPad or iPod touch in iTunes on PC

Update software on iPhone, iPad or iPod touch in iTunes on PC – Apple Support (UK)

If you have a Mac see this Apple article: Update your iPhone, iPad, or iPod touch with the Finder

Update your iPhone, iPad, or iPod touch with the Finder - Apple Support

5.

If they are On, turn Off Hotspot & VPN.