Not sure if this is the right sub but I'll try anyway.
This source claims to be a recording of captain Arthur H. Rostron, yet it doesn't give a year or any info about what it was recorded for or by who, the description simply says:" I am not fully certain as to the date of this recording". I couldn't find any other mentions of him ever giving an interview on the radio or any other form of recording, so if you happen to have any info verifying or disproving the validity please let me know.
https://www.youtube.com/watch?v=8wbtcEuef_c&t=1s
The link you have posted is to a reading of the chapter on the Titanic that Rostron published in his autobiography, Home From the Sea (1931), as you can verify by comparing the words on the soundtrack to the original. So it's certainly not an interview, nor a spontaneous recollection.
Like you I have not been able to find any suggestion that the voice on the recording is that of Rostron. Rostron actually has an IMDb page, which records his appearance in some old silent newsreels, but does not mention any audio recording.
It is certainly not impossible that Rostron made a recording of himself reading the chapter before his death in 1940, but I'd expect the fact that he did to be more easily recoverable if this is indeed his voice. Furthermore, an annotated bibliography of Titanic-related material suggests that the Titanic Historical Society produced a two-cassette audio edition of Rostron's original account in 1998, with an introduction by Rostron's daughter. That may be the original source of the recording posted to YouTube, though the voice on the recording certainly doesn't sound very contemporary.