iPad perpetually out of storage space

You actually have an older 16 GBs iPad.

A 16 GBs storage iPad starts out, after initial drive formatting, at approx. 14.5 GBs of free, internal storage.

After the installation of iPadOS, at approx. 6 GBs, leaves 8.5 GBs of of free, internal data storage space.

iOS/iPadOS needs a minimum of 2-4 GBs of constant remaining storage, AT ALL TIMES to insure/ensure nominal, acceptable operation of all iOS/iPadOS functionalities.

This leaves a 16 GBs iPad with ONLY a practical, usable storage space of approx. between 5 and 7 GBs AND THAT IS IT!

This is virtually NO internal data storage to do much of anything with that iPad!

This is why Apple quietly, silently did away with all 16 GBs iOS device back sometime in 2016.

Whatever you have actually tried eludes your description. Without wishing to go over ground that may already have been covered,

Without wishing to go over ground that may already have been covered, if your internal storage is full, your only option is to better manage the storage that you have - and delete unused Apps or unneeded data. 

These support pages may be helpful:

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

Manage your iCloud storage – Apple Support

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

Clear Other storage on your iPhone or iPad - Apple Support

iCloud storage is a supplemental, secondary, online, offsite pseudo data storage and data sync-ing and data backup service.

iCloud does NOT magically give your iPad more internal data storage.

Any iOS device has an internal flash memory data storage drive of a fixed amount that cannot never, EVER be increased!

If you need additional internal iPad data storage there are ONLY two things to do.

Purchase a larger data storage capacity iPad OR backup, then remove stuff off of your existing iPad.

You need to free up free data storage space on an iPad/iDevice.

You need to start by backing up your iPad to either Apple's iCloud ( you may need to pay for additional, monthly storage space ) or by connecting your iDevice to a computer running the latest iTunes or, if a newer Mac with a much newer macOS, use the macOS Finder to do the device backup and/or some other offsite, “cloud” data storage/backup services, like Google Drive/Photos, BOX, DropBox, Amazon Cloud Drive, Flickr ( for photos/images ), etc.

Then, delete any installed apps that you no longer use or use infrequently enough that they aren't missed (you can always reinstalled any purchased or free apps at some time later, as you need them).

In addition, uninstall any games that appear to be using a sizable chunk of your iPad's internal storage, as well!

( A disclaimer to the immediate advice above. )

If you own an nearly 8-9-year old to nearly 10-year old iPad, you SHOULD ONLY delete apps that you have ever seldom used or no longer use as there is NO WAY now, in 2020, to retrieve these old apps from the iOS App Store, any longer.

( Third party iOS app developers have permanently removed all of their older, legacy apps from Apple's iOS App Store servers forever!

There are, VIRTUALLY, NO more older, useful, popular, third party apps for over 8-10-year old iOS devices, any longer! )

Then, thin out/delete any music/videos/movies/TV shows/podcasts/screen recordings/images/eBooks/

eMagazine/iBooks/audio books/Documents/PDFs, etc.

If you use a POP email account on your iPad, delete/thin out any old saved emails that maybe saved to your iPad.

To permanently delete photos/images off of your iDevice go into the Photos app and find the Recently Deleted folder/album and permanently delete any photos/images stored in that folder/album.

Also, you can check to see what other stuff is eating up all your valuable data storage space by going into the Settings App,,General Settings Panel, in the right hand column, tap Storage & iCloud usage panel ( in iOS 11 and later, it’s now called iPad Storage ), in the top Storage Panel, tap the Manage Storage panel, wait for a list of apps to generate that will tell you how much space each app is taking up on your iDevice.

Then, for older iPads or iPads with a Lightning connector, do a hard reset of your iPad by holding down both the Home and sleep/wake buttons simultaneously until your iPad goes to black and restarts with Apple logo, then release the buttons.

iOS devices and iOS/iPadOS need to maintain, at the very least, a minimum of between 2-4 GBs of free data storage space, AT ALL TIMES, to insure normal/nominal functionalities/operation and general overall performance of any iOS device running iOS/iPadOS.

Good Luck to You!

Well,

With only approx. 6 GBs of usable internal data storage space, small background storages of data maybe quickly filling up what little space your iPad has.

You need to perhaps make sure background app refreshing is shutdown globally for whatever apps are left on your iPad.

You can turn background app refersh OFF, globally, for the entire iPad in the Settings app, General, Background App Refresh.

See if that helps, at all.