Update
I have found that CLC stays DO count for points and DO count for stay credits at Hyatt Places. I have not tried it anywhere else but CLC believes they count in all programs.
Becareful on how you pay for the room(s). Your CLC is tied to a Mastercard which may or may not be your preferred card. If they bill to CLC your charge will show up on that card. In our case it is an obscure card we seldom use so we were surprised to see charges on there.
Present the card you want to pay the room with along with your CLC and make sure the clerk swipes it and bills the room to it. I have learned from experience that not doing so can mess up your bookkeeping.
There is a Favorite list you can create on the CLC web site so you can quickly find your desired hotel and see the current rates.
Safe Travels