I was able to find a solution on my own. I compared the Juniper's output in the session_log between netmiko 2.4.1 (which does grab the timestamp on command output after you do 'set cli timestamp') to netmiko 4.4.0 (which was not grabbing the timestamp)

...I found that nemiko 4.4.0 was sending 'set cli complete-on-space off' on initial login. So I made my script do 'set cli complete-on-space' in addition to 'set cli timetamp' , and then netmiko started obtaining the timestamp on command outputs.

I know I'm answering my own question, but hopefully this helps others who might run into this.