On migrating pages from SharePoint 2013 on-premise to SharePoint Online, does the migration tools like ShareGate, Metalogix or Microsoft own tool - SharePoint Migration Tool migrate the pages along with OOB web parts over it (like Content Editor Web Part or Script Editor Web Part)?
We use Sharegate and I've done several migrations from 2010 & 2013 on-prem to SP Online. It has migrated every OOTB web part EXCEPT Summary Link Web Parts. I've discussed this with their support and there's no plans to fix what I consider a gaping hole. We use SLWP to make pages easier to edit for the average SharePoint user and because it's flexible in its display options without a lot of custom CSS/JS. Based on what they said, it has something to do with a list item null reference because SLWP are stored as lists in the SP db.
In order to compensate for this we've built a custom migration app that copies just the SLWP from source pages to destination pages. PowerShell can handle the SLWP migration, it's just not something Sharegate wishes to add apparently.
We're also using classic Publishing sites for our intranet, not the "new experience" sites that MSFT is pushing hard.
As for the other tools, if nobody else answers you can just ask their sales folks directly, or ask for a demo. I demoed all of them a couple years ago but didn't get that far into the weeds. Sharegate serves all the basic day to day migration needs for lists, libraries, etc. at a MUCH cheaper price point.