At my wits end, cannot update from Mojave

Your USB installers are failing because you must first boot into Recovery Mode or Internet Recovery Mode to modify the security settings using the Startup Security Utility to allow booting from USB.

As for macOS not allowing to install the later upgrades....perhaps this laptop is (or was) managed by your employer (or school if a student) which may have rules in place to prevent certain versions of macOS from being installed. Sometimes installing the OS from USB will get past the restriction if it the restriction is only within macOS itself.

Have you tried booting into Safe Mode to see if you can install a later version of macOS?

FYI, if the internal SSD is erased, then you lose all ability to authenticate to the T2 security chip since you have a 2018+ model. I'm not sure what happens to the security setting for booting from USB if it was previously enabled. When this happens, your only option may be Internet Recovery Mode.

If enabling booting from USB doesn't allow you to boot & install later versions of macOS, then try Internet Recovery Mode using Command + Option + R to attempt to access the macOS 13.x Ventura online installer. Unfortunately some Macs may only boot to the online installer for the OS which originally shipped with the Mac from the factory no matter the keys used for booting. In that case you may have a problem since for the last few weeks nobody AFAIK has been able to install older versions of macOS from Internet Recovery Mode due to some sort of server issue (or Apple is no longer supporting those older online installers...we don't know since there has been silence from Apple...some regular contributors have already mentioned the issue to Apple behind the scenes). In that case you can try a firmware "Restore" to reset the T2 security chip & firmware to hopefully allow the laptop to access the Ventura online installer.

The only thing that comes to mind is this:

MacOS Monterey Won’t Install Onto Macs wi… - Apple Community

If that Mac was upgraded with a third-party drive, and you were trying to install Monterey, then it sounds as though Monterey would refuse to install a system firmware update that your Mac needed. You'd need to install Monterey on an Apple drive (to get it to install the firmware), and then after that, you could swap in the third-party drive, and install Monterey on it. Or that's what I gather from the thread.

But you say that it happens with Catalina and Big Sur - not just with Ventura. Also, I don't know whether your Mac has an Apple-supplied drive or a third-party drive. It doesn't sound like this could be it.

Still, it's the only thing I can think of that's even close.

marksy68 wrote:

I took the laptop to an Apple Certified store and they couldn't get it to upgrade either. The tech guys says he's never seen this.

I would love to try the firmware "restore" you mentioned as I have not tried that. But I don't have another Mac to do so. I will go searching.

The AASP should have tried a firmware "restore" as well. Apple or an AASP should be able to try that for you. Technically the "Revive" option should be all that would be needed, but hard to say.

I have heard of some people using the display models to assist them sometimes, but you would first need to download the Apple Configurator app from the App Store on your own system first, transfer it to USB stick, then to the host laptop....just make sure to delete it from the host Mac afterwards & remember to empty the Trash on the host Mac as well since items downloaded from the App Store will retain your AppleID which would be easy for all to see if the app is left behind on a Mac you don't control.

I'm so glad the "Revive" worked for you!

Thanks for following up with the results.

You're welcome.