I also had a problem where I could not move components into and sub-assembly component in the browser - but I since found a solution that worked. It seems the critical issue is that I created the sub-assembly AFTER creating the components, when I later realized I needed some sub-assemblies. The solution is to delete that sub-assembly, wind back the timeline to well before creation of the components and then create the sub-assembly early in the time line. Then when you wind the timeline forward, you can then move all those components into the sub-assembly. And the joints don't get disrupted either which is a relief. This has solved what has been a really frustrating issue. Another benefit is that it also gets the sub-assembly higher up in the browser instead of way down the bottom. This method is also good for managing fasteners and it sometimes makes a lot of sense for me to create a "fastener kit" sub-assembly and put all the many McMaster, nuts, bolts etc in there so that the browser list does not get so long and cluttered.
Having said that, the McMaster-Carr parts download thing is great but their part descriptions that come into the Fusion auto-parts lists are terrible. Either Autodesk or McMaster needs to fix that. Because of that I no longer use the Fusion auto parts list functionality on my drawings, and instead just create a custom table on my drawings and manually enter the parts so that I can more quickly control the parts descriptions etc more directly. If they ever fix the descriptions I suggest they adopt Spanish kind of grammar for all parts, which is; NOUN, adjective 1, adjective 2, adjective 3, etc, etc. Having managed thousands of elevator parts for many years I can confirm that this is by far the best naming convention