HD Rifle
Saiga is good, and in .223 the wide range of ammo available gives flexibility.
Kel-tec SU-16B (16C if you're not in CA) is reliable, compact, light. Eats anything in .223/5.56 and just keeps shooting. Takes standard AR mags. Right at $500, but you can find them for less.
Marlin 336C in .30-30 will certainly git-er-done, but ammo cost will make it less fun to do regular target practice. Also makes a competent game rifle all the way up to black bear and large deer (elk, too, if you place your shot).
Marlin 1984C in .357/.38 special would be a good choice, and the range of ammo available in .38 special and .357 magnum is substantial.
If varmints are likely to be a factor, then any of these would be fine, but I would still lean toward the Marlin 1894C, as .38 special is a reasonable pest load, and .357 will take deer-sized game if needed.
With practice, a guy can put lead downrange from a lever gun with speed and authority, addressing multiple targets. The key is practice.
If getting enough practice is an issue, then the autoloaders like Saiga and Kel-tec will still allow a decent practical rate of fire.
To address the "Mini 14s suck and can't hit a barn at 50 yards" issue: please be aware that Ruger has scrapped their old barrel equipment and the new Mini 14s are target quality.
I don't know what range of serial numbers kicks off this new quality version of the rifle, but it's definitely different.
Oh, and that breaks the $500 rule, since you aren't likely to see one of the new make floating around the used market.