Unreal for the UI is correct, that's what runs the menus before launching the specific game.
The games themselves are running natively - genuine PC ports. The first 2 games actually had PC editions back in the old days, and those versions are what's used for the MCC release. The others were built for Xbox 360, which isn't all that different to a Windows PC, and porting them was practical.
There are some big differences between old Halos and this. First, yes, this is quite badly optimised, and old Halos had to be very optimised so they could run on old consoles. But it's more than that. Despite being pretty small in scale, this game uses a big chunk of tech that old Halos just couldn't. Halo Reach is not running 4k textures, or soft body physics, or high-detail hair physics, or advanced lighting with subsurface skin details, or advanced fluid physics. Those are all things that are very performance intensive, even today.