Why won't my Xbox Series S connect to internet extenders?

Extenders are notoriously unreliable. Mesh systems like EERO are much much more reliable.

You'd have to ask tech support eith who makes your extenders.

I would recommend looking at power line networking which is much cheaper than mesh, and typically works well. Or just run an ethernet cable.

Rewards app won't work for either the Xbox One S or the Xbox Series S trying multiple internet connections and DNS servers

Okay,

This is a public network, correct?

This is a common issue for many players that require an authentication page.

Please try this.

Open Settings on Xbox.

Select the General tab.

Click on Power Options.

Under the “Options” section, click the “Shutdown now” option.

Click the “Shut down” button.

Once done, power up the console and see if the issue continues.

I have, but it still doesn't want to connect to it.

Okay,

That is good.

Are you getting any specific errors or is it simply not allowing you to connect to that specific network?

As a temporary test, have you tried connecting to a different network such as a mobile hotspot?