Yeah, HASL can be pretty lumpy, heavily depends on the manufacturer, so I usually use it only for hand-soldered prototypes, provided there is no fine-pitch or BGA stuff. For any kind of automated assembly, I don't like it. YMMV.
OSP is a non-conductive protective layer, contrary to HASL or ENIG. It does alright for protecting copper, but has to be vaporized during reflow. It thus doesn't give the best finish surface for soldering, and leaves residues that have to be cleaned, even if using no-clean fluxes. I would avoid it unless you had to optimize cost to a high degree and it was ok with your requirements. As already said above, this is actually a cheaper finish, and if it's ever more expensive at a given manufacturer, it's because it's not their standard process, and because you are ordering only low quantities.
Just look at PCBWay for instance, you'll confirm this. If ordering 5-10 pieces, OSP is more expensive, but for larger quantities, it becomes the exact same price as HASL, and past a certain quantity, you can negotiate prices and it will get less expensive.
Additional note is that while it doesn't leave a great surface for soldering, it tends to be a lot less "lumpy" than HASL, so I guess that it might be better for fine-pitch SMD assembly and a low-cost alternative to ENIG, as long as you know the downsides.