2 added OPs extra information
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The Xbox One console is not directly compatible with Xbox 360 controllers.

However; as you have experienced, Windows 10 will still register Xbox 360 controllers, for the purpose of Xbox One streaming.

I have a USB Xbox 360 controller that can not directly connect to my console, but works perfectly fine when streaming to my PC.

As it appears, OP has attempted the same with the wireless converter, with no luck.

The Xbox One console is not directly compatible with Xbox 360 controllers.

However; as you have experienced, Windows 10 will still register Xbox 360 controllers, for the purpose of Xbox One streaming.

I have a USB Xbox 360 controller that can not directly connect to my console, but works perfectly fine when streaming to my PC.

The Xbox One console is not directly compatible with Xbox 360 controllers.

However; as you have experienced, Windows 10 will still register Xbox 360 controllers, for the purpose of Xbox One streaming.

I have a USB Xbox 360 controller that can not directly connect to my console, but works perfectly fine when streaming to my PC.

As it appears, OP has attempted the same with the wireless converter, with no luck.

1
source | link

The Xbox One console is not directly compatible with Xbox 360 controllers.

However; as you have experienced, Windows 10 will still register Xbox 360 controllers, for the purpose of Xbox One streaming.

I have a USB Xbox 360 controller that can not directly connect to my console, but works perfectly fine when streaming to my PC.