Is Ubuntu 20.04 LTS still supported for five years if I change the desktop environment?

Packages included on the Ubuntu Desktop ISO come with 5 years of support; as to packages from 'restricted', but packages from 'universe' being community supported which include flavor desktops such as XFCE & MATE only have 3 years of support, so adding extra desktops will mean your system is partially supported after 2023-April (3 years from initial 20.04 release).

eg.

On a QA-test install of Lubuntu focal daily performed yesterday (ie. what will be released as the 20.04.3 ISO) I get

guiverc@dc7700-2004:~$ ubuntu-security-status
1773 packages installed, of which:
1252 receive package updates with LTS until 4/2025
 521 could receive security updates with ESM Apps until 4/2030

Enable Extended Security Maintenance (ESM Apps) to get 1 security
update (so far) and enable coverage of 521 packages.

This machine is not attached to an Ubuntu Advantage subscription.
See https://ubuntu.com/advantage

This shows the packages that have 5 years of support, and those that can be extended via the use of ESM.

Ubuntu does not use dist-upgrade to release-upgrade for a number of reasons (to improve stability & lessen the risk of problems primarily, yes that is part of the process but only part) so much of your question applies to off-topic systems.

If you read the release notes of the release you'll upgrade into (eg. Ubuntu 18.04 LTS) you'll find instructions on how to upgrade & keep your Ubuntu system secure.

Flavors release-upgrade the same as main Ubuntu; with very rare caveats, eg. Lubuntu 18.04 LTS was the last with LXDE thus upgrade was supported via re-install, Ubuntu Studio 20.04 LTS was the last using XFCE thus... etc but all those details are made clear in the release notes.

Both Ubuntu (Gnome) and Ubuntu MATE are made up of packages, and indeed share many of the same packages.

• Ubuntu (Gnome) 20.04 packages are supported for 5 years.
• Ubuntu MATE-specific packages are supported for 3 years.

If you want to do a release-upgrade (not dist-upgrade -- they are different), apt will attempt to upgrade ALL the packages that you have installed. If you still have Ubuntu (Gnome) packages installed, whether you use the Gnome Desktop or not, those packages will be upgraded.

• Dist-upgrades are available, but not supported. Release-upgrades (do-release-upgrade) are supported in Ubuntu. If you break your system by mis-using dist-upgrade, we will merely suggest that you re-install.

A "better" choice is a matter of opinion and personal preferences. We don't know you, so we don't know what is "better" for you.

There's a lot of hearsay and speculation in your question that I'm not going to bother to address beyond this advice: Since you are using a "customized version of Ubuntu," keep notes on how to duplicate your customization on a new install. ALWAYS be prepared to clean-install. Have backups, notes, and install media before beginning a release-upgrade. In 16 years, I've had a botched release-upgrade exactly once...and that was my own fault. However, perhaps I do things differently than you do.