theoretically this is a job for branded zones too, BSD actually solved this problem better a long time ago! just write a shim that hooks whatever linux layer to the kernel layer.
smartOS and proxmox both fill that niche for "hyperv, but it's application containers" niche perfectly. Or full VMs if you want too!
but for both the "making docker work with BSD" and "making BSD work with docker" approaches, the question is how much time a company will spend on an alternative, upstart tech for something as foundational as their container deployments, without a really compelling reason. the answer is likely "zero" and justifiably so. it doesn't matter that you can run a linux vm, it won't matter that zones work with docker or that docker can use OCI. nobody wants to discover a whole new set of bugs and footguns with a niche technology stack.
netflix really ran with it, but even they did not put application servers/containers on BSD, they just used it as a NAS store. which is fine, and that's something that can be encapsulated without letting BSD escape into the rest of your ecosystem. but even though the linux ecosystem is in a state of constant meltdown and rebirth, it's at least the enemy you know.
I wonder if that will change now that redhat has slammed the door on non-commercial users. like where do you even go after that? ubuntu is clownshoes (and the snapd stuff has offended a lot of users). Debian I guess, or OpenSUSE. But SUSE is quite niche as well. Debian is probably the "sufficiently boring" answer, but freebsd could be a good alternative too if places gave it a chance.
as I said it's very impressive how boring freebsd is. Most likely the thing you are trying to do will be either building/loading some module in /etc/rc.conf, or editing some config file in /etc/, or copying some template from /usr/local. And you just do it, there's no surprises, and you move on.
There is some interesting work happening to get an OCI-compatible runtime working
https://github.com/samuelkarp/runj