5mbps internet should be adequate for one video stream but it looks like you are overpaying for it by a wide margin. That plan should be about $40/mo for unlimited or at least 200Gb of data. For $68/mo you should be getting at least 30Mbps with unlimited data plus a much newer modem/router.
If the TSN app doesn't offer Chromecast casting then it's using screen mirroring. That will be much less satisfactory than direct Chromecast support and may be the cause of the issues. A better router could be the solution.
Not really familiar with the suffix's like G, AC, AX.
All you need to remember is that the router should support AC wifi.