The problem has nothing to do with the cache being disabled. It is that the plugin is using a different cachebase setting than the cache that saved the data.
The cache that saved the data used the site cache folder located at /cache, while the plugin is using the cache folder located at /administrator/cache. It doesn't see the data because it's looking in ...
I had some time to check this problem but I was not able to duplicate your issue. I created a short sample caching test in one of my layout files like:
$myCache = \JFactory::getCache('my_cache');
$dataToCache = "<h2>this works now</h2>";
$myCache->store($data = $dataToCache, $id = 'yep1', $group = '...