Our ski club used this display method last season. The TV is a Sunbrite Signature 43" mounted outdoors about 120 m from the timing location. We have cat 5 buried wiring connecting these points. We used a chromecast over ethernet converter because wifi was not reliable at that distance, internet is supplied directly to the ethernet cable by plugging into router. The chromecast converter is located out at the TV and is mounted at the end of the ethernet cable with connection to the chromecast dongle. We used a PoE injector and a mid-run switch to improve range over cat 5.
The procedure for use is fairly straightforward. Bring up a live timing screen on a Chrome browser page on a device in the timing room where you can monitor it, cast the screen to chromecast. It will mirror everything on the chrome browser, which means you'll need to follow any change of race file such as if you have more than one age group or gender. Most timers I know like to monitor live timing on a separate device to make sure messages are posting and times updating, so this is something you'd likely do anyway. You can use chrome browser settings to maximise screen size, and as you say, better to mount the display screen in portrait rather than landscape for this application.
However now there is a better way. This year Split Second includes a function called 'digital scoreboard' that produces a screen on the timing computer with bold graphics and already size formatted for 1080p TV. Just connect the TV display to the timing computer via HDMI and set it up to be a second monitor. Drag the digital scoreboard screen over to the second monitor and you're done, constant updating scoreboard separate from live timing and internet and the TV can stay in landscape orientation. If your distance is too great for cable connection, there are HDMI wireless transmitter/receivers to send signal to TV.