Point of order: SDDM is entirely unrelated to the KDE project.
I've been using the Breeze Dark theme for approximately forever and I've never run into the problem you're describing. However, I've very rarely used SDDM... I find its default rainbow-colored background intolerable and use LightDM instead.
Do you happen to remember configuration that you ended up having to change, and is that computer running Nvidia graphics hardware with the closed-source drivers?
I went to System Settings > Themes > Login Screen (SDDM) in KDE settings and changed from the default Breeze to Maldives, and that broke SDDM login screen to black text mode with a cursor. Later searching for fixes I found that only Breeze was compatible with Qt 6.x, and any other choices there would break SDDM the same way (I did not try it though).
First, I had to figure out how to manually mount LUKS-encrypted laptop drive while booting from a USB stick, that took a while.
Trying to recover, I re-installed kde, sddm and sdd-kcm and qt5-declarative packages. Still broken. I made sure /etc/sddm.conf was the default configuration, still broken. Then finally I stumbled upon /etc/sddm.conf.d/kde_settings.conf, which was still overriding defaults to Maldives. Deleting it finally fixed the SDDM login.
My wife was thoroughly not impressed with Linux out-of-box experience!
No Nvidia graphics, this was a Lenovo Yoga laptop with AMD graphics.
I've been using the Breeze Dark theme for approximately forever and I've never run into the problem you're describing. However, I've very rarely used SDDM... I find its default rainbow-colored background intolerable and use LightDM instead.
Do you happen to remember configuration that you ended up having to change, and is that computer running Nvidia graphics hardware with the closed-source drivers?