Do all Tiki installations need to go in the 'public html' folder?
First, if your Tiki is published then it needs to be in a folder accessible through Apache2. (that’s why it is often public_html or htdoc or html... the default settings)
To answer a little widely and IMO, it depends of "external" factors (for what your Tiki is used and your IT Admin skills)
If this is intranet or your Tiki(s), meaning you of very few people use them, you can create inside public_html as many as Tiki(s) (subfolders) as you want and all should work fine (create corresponding files_gallery folders, etc).
To land on one of them, add the path to your domain. IE: https://www.mydomain.com/tiki21, https://www.mydomain.com/tiki22 etc.
If you know your way through server administration and how to set vhosts, then you redirect domain to different path than the default and set them part of the www-data group.
BUT I had to say that many things (libraries, software, packages, etc) work by default using Apache2 defaults and it can be a nightmare to adjust options and have them permanent (not reseted on server reboot).
And, if this is a public website you want it to be directly accessible with minimum of directories and navigation (https://www.mydomain.com) or eventually https://tiki21.domain.com
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