Yes I am sure that the website does work on your phone and computer (as it does on my own devices), but those are not running webOS. Just because something works on one browser does not mean it will work in another. Did you ask MotoGP if they had tested their browser in webOS or if they just tested it on mobile & desktop platforms? Websites have to be written to be compatible with the browser trying to serve the page, for instance Internet Explorer would need several different pieces of coding to be viewable in all the different releases. As I have written else where in this thread the webOS browser probably does not recognise the JavaScript being uses to display the video, and it should be relatively easy for MotoGP to engineer their site firstly to recognise the webOS browser and secondly to serve it with a version of JavaScript that it can understand and display.
http://webostv.developer.lge.com/discover/discover-webos-tv/
It is easier for a website to write a site to serve the correct JavaScript to a web browser than for a browser manufacture to cater for every esoteric piece of coding language out there. It should not be difficult as webOS's browser is based around WebKit as used by Safari, Opera, Chrome and even now Microsoft's upcoming releases of Edge. Most websites are not catering to webOS or even Smart TVs in general as they still make up a very small percentage of devices browsing the web, all sites need to be told to look for specific browser ID strings:
https://developers.whatismybrowser.com/useragents/explore/operating_system_name/webos/