The only Display Manager I know of that supports video backgrounds OOTB, is SDDM
It works (as evidenced by this video), but is not simple to install.
The basic process requires QT 5, and the code needs to be compiled.
Quote from the installation documentation:
SDDM uses CMake to configure the build and it can be compiled with the
typical cmake build process:
mkdir build
cd build
cmake .. -DCMAKE_INSTALL_PREFIX=/usr
make
After successfully building the software, type following to install.
sudo make install
When found, systemd will be used for power management. When systemd is
not available build will fallback to using upower. If upower can not
be found either, shutdown/reboot/suspend/hibernate actions won't be
available. Note that, when systemd found, HaltCommand and
RebootCommand config file entries has no effect.