core: frontend.set_theme doesn't change default theme modes

The problem

service: frontend.set_theme
data:
  name: default
  mode: dark

should change current (=default) theme mode to dark, but it doesn’t work - nothing happens (it doesn’t work neither with mode: light)

What version of Home Assistant Core has the issue?

core-2023.12.3

What was the last working version of Home Assistant Core?

No response

What type of installation are you running?

Home Assistant OS

Integration causing the issue

No response

Link to integration documentation on our website

https://www.home-assistant.io/integrations/frontend/#defining-themes

Diagnostics information

No response

Example YAML snippet

No response

Anything in the logs that might be useful for us?

No response

Additional information

https://community.home-assistant.io/t/change-dark-light-mode-of-default-theme-based-on-automation/516181/2 https://community.home-assistant.io/t/day-night-theme-switching-automation/24887/17 https://community.home-assistant.io/t/set-default-light-dark-lovelace-theme/412486