Well, the original idea behind "skinny bundles" like Sling was not to include local channels. The idea was instead to supplement local channels (received via antenna) with traditional cable channels delivered via the Internet.
So when I first subscribed to the Sling Blue package, I was surprised that Sling did have a few of my local stations: IIRC they had the local NBC, Fox, Univision, and Unimas affiliates. No AirTV required (it didn't even exist yet). I suspect Sling Orange might have had ABC. (You can subscribe to both, but that costs $40/mo. vs. the typical $25/mo. for just one package.) It's nowhere near a complete local channel package, but I wasn't expecting any local channels at all.
I assumed those channels could be recorded via Sling's "cloud DVR" just like any other Sling channels, but since I never subscribed to Sling's cloud DVR I never could try it.
If you're looking to get both local and cable channels over the Internet and have no interest in receiving anything with an antenna, Sling TV probably isn't your best choice, unless you're interested in only the few OTA channels they happen to include. Instead, you might check out your local cable company. Mine (Spectrum) offers any stations you subscribe to, including many shows on-demand, via Internet; you don't even need a cable box.
However, Spectrum doesn't provide a "cloud" DVR; if you want to record, you'd either need one of theirs or one that accepts a CableCard (i.e., TiVo). If you don't want a DVR but on-demand offerings aren't good enough, PS Vue may be about the only choice that provides both local channels via Internet and a cloud DVR.