Q:

Does Cross Site Publishing impact the ability to Save Sites as Templates

It's well knowing that you can't save a publishing site as a template.  The moment you turn on publishing you lose this capability.
Does activating Cross-Site Publishing impact the ability to Save a Site as a Template?
Thanks!

A:

If you plan to use Cross-Site Collection Publishing for Publishing Site, So
Yes, Working with (Not just Activating) the Cross-Site Collection Publishing for Publishing Site Will impact the ability to Save a Site as a Template functionality.

Note
Publishing site collections for the publishing site must have the SharePoint Server
  publishing features activated so you must have 

A Publishing Site Collection template (Recommended) or 
A site collection template that has SharePoint Server publishing features enabled.

That means you will lose the functionality of Save Site As
 Template

For more details, check Plan publishing sites for cross-site publishing in SharePoint Server

If you plan to use Cross-Site Collection Publishing for Authoring Site, So
It depends, if you need Approval workflows and scheduling, you must also activate publishing features that lead to losing the functionality of Save Site As
 Template. 
Otherwise, activating Cross-Site Collection Publishing feature doesn't impact the ability to Save a Site as a Template.

Note: Just activating the Cross-Site Collection Publishing doesn't impact the ability to Save a Site as a Template
  functionality in case,

You don't need to use Cross-Site Collection Publishing for a Publishing site.
You are using Cross-Site Collection Publishing for Authoring site and you don't need publishing features like Approval workflows and scheduling.

For more details, check Plan authoring sites for cross-site publishing in SharePoint Server

For more details check 

Overview of cross-site publishing in SharePoint Server
Missing Save Site as a template 

