The Team site carries a higher number of web parts compared to the Communication site for example the Planner web part. Is it by design or is there something I need to activate to have the planner web part in the communication site?
This is indeed by design.
The extra webparts that you are referring to are called as connector webparts. They are only available on O365 group connected team sites and not communications sites or other sites.
For example, Planner webpart depends on O365 group capabilites. They are quite tightly integrated because of which I guess its not available to other site templates
Reference - Use the Connector web part