But will this species eventually die out because males cannot attract a mate or can scientist use play back recordings (of song) to teach these young males how to sing?
The article you reference is behind a paywall, but I believe I've found the study they're referencing anyhow. You can find it here for free.
To directly answer your question: no one can really answer that right now. That's not to undermine your question. Any population can go extinct when put in certain conditions.
The authors of the paper call for more controlled research,
Experiments in captivity that replicate a range of demographic scenarios could help improve our understanding of the process of cultural song erosion and its impacts on fitness and population growth .
I know that's rather unsatisfying, but it's straight from the horse's mouth. That's science for you. But I'll try to entertain you for a bit, because what would a good answer be without a bit of supposition? As I'm sure you've read, it isn't that the birds aren't singing, it's that they're not singing the right song. The authors note
Regent honeyeaters exhibit strong conspecific attraction, once roaming in flocks of hundreds . However, at least four of the interspecific singers were more than 100 km from the nearest known male at the time they were detected.
Which means when they sing the right song, they do a darn good job of recognizing each other. But unfortunately, these guys are too far apart, and as their population density decreases, they're more likely to learn songs from interspecifics (other species). The authors also measured the ability for birds in captivity to learn their own song and they suspect it could cause problems upon release. A scientist teaching a bird to sing might not be a solution either. Sure, we could just stick a speaker out in the woods and let it go all day, but it's kind of like putting a bandage on a sinking ship.
The real issue here is conservation of these guys' habitats, which involves a variety of processes beyond a boombox in the woods. There are tons of questions we still need answers to. For example, if a population decrease results in young birds learning the wrong songs, what causes the population to decrease in the first place? It's a sort of chicken or the egg question. Or, what if we just take the birds and combine their populations? Well, that might be more harm than good. What causes these birds to learn the songs of other species? Are they too smart for their own good? How heritable are the genes that encode these important songs? So on and so forth, you understand.
For more about conservation, and what can be done, check out an example of the Florida panther and what's been done to protect them. Of course, every group is unique in their needs, but just to get an idea of what it takes to conserve a species.
Great literature review from @BrownBag of the original paper. A notable addition to whether the speakers will be effective in teaching the young males the "correct" song brings in the topic of acoustic propagation.
Even if the speakers are placed in the areas with the most young males /and/ are "singing" the correct song, it does not mean all of the song's important features will projecting equally throughout the environment where the young males would be positioned. Consider these three examples:
If air temperature is higher to the East than to the West
There is denser foliage to the North vs the South
There is a highway in the NE corner of the habitat.
All the frequencies of the "correct" song will still not propagation equally under these conditions.
Overall: lower frequencies propagate further than higher frequencies, so a young male further from the speaker may only receive the low frequencies of the song and therefore still learn a low-frequency-only "wrong" version of it.
In terms of the specific examples 1-3 above:
1A) Sound waves travel faster in cooler temperatures, so young males towards the West will have a better chance of hearing the song at further distances to the speakers.
2A) Denser foliage reduces propagation distance by absorbing and scattering sound waves, so young males listening through the leaves (literally) towards the North may learn a diluted version of the "correct" song.
3A) Highway noise masks low frequencies more than high frequencies, so young males positioned closer to the highway (in the NE corner of the habitat) may learn a high-frequency-only "wrong" version of the song.
All these environmental limitations to propagation can shape song over time. They are also not predictable from day to day, so the ideal placement of a speaker could change from day to day.
In terms of time scales of evolution, it is likely that the Regent honeyeater has been able to adapt to temperature and foliage differences over millennia (since they are not extinct yet). But the human-driven effects of masking have occurred on a faster time scale (over decades, not millennia). Therefore, the Regent honeyeater may or may not be able to adapt its song fast enough under the faster anthropogenic pressure and could go extinct as a result.