No app needed. Safari handles the whole thing: convert on this site, find the MP3 in the Files app, play it or move it to your library. Four steps.
In the YouTube app, tap Share under the video, then Copy link.
Open Safari, go to our converter, paste the link, tap Convert, then Download when it appears. Safari may ask where to save; choose Downloads.
Open the Files app, tap Browse, then Downloads (under On My iPhone or iCloud Drive). Your MP3 is there.
Tap the file in Files to play it directly. To add it to your Music library permanently, transfer it to a Mac or PC and import it into the Music app (it syncs back to your iPhone).