There is no browser available for any Roku device. Since that is the only way to access your Serviio from the Internet (using MediaBrowser), the simplest method is to use a computer connected to your TV, as you mentioned.
There is one other way to access your local media over the Internet with a Roku device, but it doesn't use Serviio. Plex offers such an ability, and there is a Plex app (channel) available. You can run Plex and Serviio on the same computer and access the same media (I have done it for years). They use different ports, so you just setup your router to forward both ports and it works fine.
Remember that many home Internet connections are asynchronous, which means your outgoing speed is usually far lower than your incoming. This is fine for bringing things into your network, but not so great to send a video stream out. My DSL is 40-50 Mbps in, but only 5 Mbps out. It still works, but I have to set my video quality far lower than I see from any player at home. Often SD is the best I can get when streaming to a hotel TV with my Roku Stick. Of course, that could have something to do with the poor Internet speed offered at most hotels!
EDIT: I do see that there is now a brand new browser offered. I have my doubts that it will work with MediaBrowser, but it might be worth a shot. It doesn't say there's a charge, but perhaps the charge appears after you install it. https://channelstore.roku.com/details/56529/web
. Yep, I just tried installing and it warned me about the monthly charge. For your purposes, I would still suggest Plex.
Panasonic ST30 Plasma, Samsung JU7100 4K TV, Samsung UN24H4500 TV, Sony BDP-S3500, Insignia Roku TV, Roku 2 XS, 4, Ultra and Stick, Yamaha RX-V679 AVR.
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