Most pages aren't caching

I just noticed that WordPress is trying to set a cookie, too:

Set-Cookie: PHPSESSID=d6413d0d52f0c9a195933859b57da2b8; path=/

You can fix the code that sends those headers or you can still force Varnish to cache, irrespective of those headers and cookie, by setting beresp.ttl in vcl_fetch.

Is it a good idea to do that, though?

Also, this is happening on one WordPress site but not another. What can explain this?

And when I did a grep on “PHPSESSID” nothing came back except a module that I deactivated. What the heck is setting that cookie?

Is it a good idea to do that, though?

Depends on the site.

Also, this is happening on one WordPress site but not another. What can explain this?

And when I did a grep on “PHPSESSID” nothing came back except a module that I deactivated. What the heck is setting that cookie?

You may want to deactivate all the plugins and switch to a default theme. From there, you may activate one at a time to see what creates the cookie and the headers.