Why does it keep disconnecting the party?

what type of nat do you have?

So why does the party chat disconnect us as well?

Hi, I hope you're doing great out there. I am an Independent advisor and I am here to help you in regard to your concern.

It can be an issue with your network connection itself, Wyatt. Please perform an extended power cycle and reset your network hardware by following the steps below:

1. Unplug the power cable from the back of the router, modem, or gateway for 5 minutes. If you have a router and a modem, unplug the power cable from both devices.

2. Press and hold the Xbox button on your console for 10 seconds or until it turns off then wait for 2 minutes before turning it on again.

3. After 5 minutes, plug the modem or gateway in first, and wait for all the lights to return to their normal state.

4. If you are using a router, plug in the router and wait for all the lights to return to their normal state.

Does your internet disconnect and reconnect often? It could be one of the reasons why the update keeps on stopping.

No, it's not the same cable actually... but the same thing happens when using it wirelessly.

Just out of curiosity, what should I change in the router settings to make it wait more time before dropping a connection as a result of considering it dead (since there is no option to make the xbox keep the connection alive)?

Check this article related to the keep connection alive thing:
www.knowswhy.com/why-does-xbox-live-keep-disconnecting