Change dark/light mode of default theme (based on automation)?

Sure. Here you go.
This is a node that issues a service call to switch all browser_mod registered instances to dark:

[{"id":"31c5da83d2e68d9f","type":"api-call-service","z":"2cbe74495b5313c9","name":"Activate Dark Mode using Browser_Mod","server":"181ffc96.b00263","version":5,"debugenabled":false,"domain":"browser_mod","service":"set_theme","areaId":[],"deviceId":[],"entityId":[],"data":"{\"dark\":\"dark\"}","dataType":"json","mergeContext":"","mustacheAltTags":false,"outputProperties":[],"queue":"none","x":600,"y":2340,"wires":[[]]},{"id":"181ffc96.b00263","type":"server","name":"Home Assistant","addon":true}]

while browser_mod is a workaround, its not a solution.
I think this issue fits in here:

As of 2024.7 you can’t use a combined theme with

themename:
  modes:
    light:
      ...
     dark:
      ...

to switch with frontend.set_theme between light and dark mode.
You have to use two separate themes, one colored light, the other one colored dark to make it switch via automation. (and have your profile set to use default theme/backend selected)

I tried that, as the map-card didn’t switch between light and dark - as it uses the mode setting to show light or dark map…