Hey Tor,
the HDMI port of a Raspberry Pi 3B can easily be made to break. When I opened the Onefinity Controller for the first time, I forgot that the power switch does not interrupt power to the internal power supply (!) and I opened the wrong end. One of the wires with lacking terminal insulation touched the case and I had a Raspberry Pi 3B with HDMI port no more working then (no image anymore). Controller was still usable remotely. Reinstalled the (matching outdated OS version) video firmware on the Raspberry, but it did not help. Bought a new Raspberry Pi 3B then.
When opening the Onefinity Controller, always unplug power plug before! Power switch does not interrupt power to the power supply!
When replacing the Raspberry Pi 3B, always work ESD-safe!
Image: ESD-safe work mat, grounded. Also use a ESD-safe and grounded handwrist strap.
I would check whether the new Raspberry Pi 3B works on another monitor. Any HDMI monitor (or DVI-D or DVI-I with adapter) will be suitable.
Welcome to the forum!