[error after migration to directadmin,]

You likely got this resolved, but for others that hit these errors...

For the first error, you are likely hitting public_html of cpanel vs the symlink to /home/user/domains/example.com/public_html - if Blesta detected the symlink.  Edit Settings -> System tab ->  "Root Web Directory" to match the correct install location.

For the directory system can not be written to, make sure the ownerships for all directories and files are to the new system user. 

[If ini_set is disabled in PHP 8.1, it prevents Blesta from working]

That would be the case in other versions of PHP if ini_set is in the disabled_functions list, correct?  (though disabling that function would break a ton of other scripts like Wordpress)  Good defensive coding doesn't hurt though. 

[Get Client ID and Service ID for Plesk and cPanel Usernames]

Just to confirm, you want to get the Blesta customer number and service id from within Cpanel and Plesk?   You would need a plugin / extension / addon for both systems to connect to blesta to request that info and  that would require a custom plugin. 

What would be the end goal for this? 

[Blesta to Cyberpanel initial connection failed]

Do both servers support ipv6 and so you have allowed access for ipv4 or ipv6 but not the other?   Though you mentioned using the IP, so that should rule that out but to be safe I would check on that. 

Are Blesta server and Cyberpanel server separate?  If not, adding 127.0.0.1 might be needed also.

And finally, can you curl to the cyberpanel url and get some sort of response? 

curl -v https://server.example.com:8443/  

That should output the page with headers and such.  If you can connect to it that way, then at least the firewall isn't blocking and is either login info or the allow list. 

[Where is the License Manager]

I believe the license manager is a paid plugin and so you have to order it first via the addons page in the Blesta dashboard.  Though looking at that, I don't see an option to order the license manager but multiple pages do indicate it is a paid plugin.

[support_manager poll_tickets Call to undefined function mailparse_msg_create()]

It is possible cron is using a different PHP install than what you're seeing on the web part.   For example, on a cpanel, plesk, or other control panel system you can multiple php versions and perhaps even different instances of the same version.  I suspect the cron is using one install while the web is using a different one.   So you need to check the cron is using the same PHP version and install as you're expecting.