Install the VLC PLAYER (it brings the right video codec) on your android device (fire tv), once installed, try your IPTV app again. Try also XCIPTV app which is free or charge, if you need help, send a pm to me and I'll guide you.
You need a seperate account:
https://iptv.community/threads/security-update-multi-connections-are-no-longer-available.19097/
PC, you can try this:
https://iptv.community/threads/tutorial-myiptv-player-windows.18171/
Fritz Box isn't bad but if you think about a new router, better watch out for routers with integrated VPN CLIENT.
And try VPN, that most likely solve the issues.
smart tv = limited cache/memory and no possibility to use vpn.
mag box is old fashion... better get an android box and you will enjoy best flexibility, you can install all apps and also vpn.
e.g. amazon fire tv stick OR xiaomi mi box.
Sounds like TV cache/memory limitation, reduce bouquet list, 2-3 groups only, remove power cable from tv (clear cache) and then try again.
If you need help, feel free to PM me.