The fact that the car does not recognise the trailer will not stop the trailer braking system.
( There are hydraulic / Air / electrical braking systems on much bigger trailers but not on standard size UK horse boxes.)

The brakes will activate due to the force applied via the shock absorber section in the trailer draw bar.
See the links below, do not know your engineering or spannering knowledge so apologise if I am 'teaching granny to suck eggs' so to speak

https://brink.eu/en-gb/towbars/safety/overrun-brake/

https://www.maplelanetrailers.com/braking-system/

The brakes could be fine, but if the draw bar section is seized, it will not apply pressure to the brakes, thus putting extra load onto the tow vehicle braking system.

Also do not forget to check the emergency wire for brake activation, in the event of trailer / tow vehicle separation.

As for the LED lights issue, if the LED bulbs are removable, change them for ones with built in resistors, fairly cheap, if it is a complete LED board then not sure how / where to fit the resistor, but a wizard on here will know how. The is usually one lurking around

Cheers Merchy, every day is a school day, excellent info, I have booked trailer in next week for brake check, makes sense when you see it like that,,,