As far as I can tell, Lutris heavily relies on WINE, multilib, Linux GOG installers and the Steam client –either the Windows or the Linux version– for modern games to work. Unfortunately OpenBSD removed support for multilib, WINE and COMPAT_LINUX long ago, so I find it extremely unlikely for something like Lutris to ever run on OpenBSD. Nevertheless have a look at the Shopping Guide for Games on BSD which contains a database of games playable on OpenBSD (and other BSDs for the vast majority), mostly by relying on emulators or the fnaify script.
NetBSD, on the other hand, fully supports multilib, has a working linux binary emulation, which is known at least to be able to run some linux GOG installers, and provides emulators/wine + wip/win64 for WoW64 support (see this netbsd blog post to learn how to get it working properly). Here's also a really good tutorial on Running Wine in a 32-bit sandbox on 64-bit NetBSD using sandboxctl.
That said, I don't know of anybody who actually tried to port/package Lutris for NetBSD. I'm not a gamer myself (if we exclude old games, emulators and foss engines), hence I'm probably not the best person to answer such a question.
FreeBSD may be ahead in this, given their linuxulator supporting more syscalls, their more mature WoW64 support, DXVK/Vulkan support and more up to date accelerated graphics drivers. Actually there's Homura, a Lutris-like wrapper built around wine and designed specifically for FreeBSD, and there's a working Steam port of the Linux Steam client using the Linux compat layer. Worth mentioning also a PlayOnLinux port for FreeBSD called PlayOnBSD.
I'm positive it's would take much to port all of this to NetBSD as well, but somebody should take the plunge and invest their time first into such a task. As of today, I see no interest in it within the community, much less among devs.