Answer is, it depends. Any controller speaking to an aircraft has a responsibility to make sure it goes where it's supposed to, as I mentioned in the other thread, whether it's a Tower controller or an Approach guy. From what I can tell, Branson's tower is a contract tower, which means any Approach control services are conducted off-site in Springfield, or by Kansas City Center. Center approach services are usually only done late at night, taking over for approach facilities that close for the night (if they're not 24 hour ops) and generally are one at a time operations, as depending on terrain and type and location of the emitter center radars are optimized for higher altitudes and don't do so well for lower altitudes. Additionally they have a much slower refresh rate, about 10 seconds per full 360ª sweep. Anyway.
If whichever facility the pilots were talking to was still open, then yeah this is very bad for whomever was working the aircraft. But from what I can see on airfield info for https://airnav.com/airport/KBBG, Tower hours are 0700-2100, which were the same as my previous tower. If the tower was closed, then most likely the frequency becomes a CTAF/UNICOM and there'd be no one on-frequency to observe the wrong track and issue any warnings or guidance. Usually at night Approach or Center frequency changes (ships) the aircraft to Tower 15 miles out or so, sometimes more esp if it's slow...and other facilities don't have ready access to neighboring facilities' frequencies, so if Approach or Center shipped the aircraft they wouldn't have a ready way of issuing correction, short of transmitting on Guard. I did see a media report yesterday that someone in ATC did attempt to warn SWA, but there was no detail (typical). Air carriers are supposed to monitor Guard, but if some guys are having full fucking conversations over Guard (happens all the time!) then it's effectively com-jammed and useless. All it takes is one or two guys to monopolize the frequency. Finally facilities do have portable, tunable radios however they're limited in range.
It all depends on what time SWA landed, I'm too lazy to look it up, but I'd say there's a decent chance BBG's tower might have been closed, and if so there's nobody to warn them. Maybe Springfield or Kansas City were trying to warn them on Guard, but obviously they didn't hear either from low altitude/poor reception, other people dominating Guard, the radios turned off, distractions in the cockpit, etc.
Finally...for places open, yes there's phraseology for an aircraft inbound that's not in sight:
CTRD means Certified Tower Radar Display.