I can see that "Register host" is designed to setup puppet in the Linux registration default template:

<% if @host.puppetmaster.present? -%>
<%= snippet 'puppetlabs_repo' %>
     <%= snippet 'puppet_setup' %>
<% end -%>

The only place the puppet setup could come from is the host group specified in Register host. If I use a host group with puppet settings these are not applied to the host and so Puppet is not installed on the host.

Attached are screenshots showing the host group with puppet settings and then the host after registering where the same settings are blank. But it seems those values on the host are not inherited from the assigned group.