a good SE got shutdown recently due to "poor business"
This isn't an accurate representation of the position of SE overlords. See the full statement.
The reason for shutdown wasn't "no profit", "low traffic" or "few questions": there are SE sites with less traffic and fewer questions asked than Productivity, yet they remain. If the call for moderators was met with at least a bit of enthusiasm, the Productivity site would still be here. See Graduation, site closure, and a clearer outlook on the health of SE sites
Chess has three moderators (appointed back in 2012) who still visit regularly: two in last week, the third visited two weeks ago. Flags get handled without unusual delays. And looking at the list of active users I see some that would likely step up, should there be a call for additional moderators.
First, there is Board and Card Games site, where chess is on topic. If this site had to shut down, I expect its content would be migrated there.
Setting up a new site on another platform is a large task but not an unimaginably huge one: the new platform could be, for example, Discourse. There are weeks between shutdown notice and execution: (Productivity had 3 weeks).
Existing content is an issue. Look for a Q&A platform able to digest SE content, including the chess notation parser. (I'm not sure one exists out of the box; one probably has to make modifications to open source Q&A software.)
Perhaps the most practical way is to set up a static site rendering the content of "Chess 2012-20xx" and start from scratch elsewhere. Jeremy Banks has an archival project for failed SE sites ("collapsed stacks"), for example Startups. If a static, browsable site exists, it can be used for references and also to draw some of the search engine traffic which can then be directed to the new site.