I think the answer boils down to what you mean by "knowing" the intervals.

To do this [sing a tune back] you surely need to be able to know the intervals in the tune you have just heard and then replay it back with your voice.

I don't think "knowing" these intervals in order to sing something back means you know if an interval is, e.g., a major or a minor third. When you sing something, you're just mimicking something you already know. There's not necessarily any true "knowledge" here; just ask the countless opera singers that sing for hours in another language without knowing how to speak it. The same is true here.

But there's an added wrinkle: intervallic ear training is problematic because it's teaching you intervals within a very specific functional context. People use "Somewhere Over the Rainbow" as an ascending octave, but it's really an ascending octave from scale-degree 1 up to another scale-degree 1. If you're singing scale-degree 3, thinking of "Somewhere Over the Rainbow" won't always be helpful; you may well sing up to scale-degree 1 (a minor sixth), not an octave up to scale-degree 3.

As another example, people can know what a major second is (it begins "Happy Birthday"), but they may only be able to recognize it accurately when it's between scale-degrees 5 and 6 in major (as it is in that tune). They'll likely have some problems hearing—and definitely creating without prior reference—that same major second between scale-degrees 3 and ♯4.

I think the answer boils down to what you mean by "knowing" the intervals.

To do this [sing a tune back] you surely need to be able to know the intervals in the tune you have just heard and then replay it back with your voice.

I don't think "knowing" these intervals in order to sing something back means you know if an interval is, e.g., a major or a minor third. When you sing something, you're just mimicking something you already know. There's not necessarily any true "knowledge" here; just ask the countless opera singers that sing for hours in another language without actually knowing how to speak it. They can reproduce the sounds, but they don't fully understand the meaning behind them. The same is true here.

But there's an added wrinkle: intervallic ear training is problematic because it's teaching you intervals within a very specific functional context. People use "Somewhere Over the Rainbow" as an ascending octave, but it's really an ascending octave from scale-degree 1 up to another scale-degree 1. If you're singing scale-degree 3, thinking of "Somewhere Over the Rainbow" won't always be helpful; you may well sing up to scale-degree 1 (a minor sixth), not an octave up to scale-degree 3.

As another example, people can know what a major second is (it begins "Happy Birthday"), but they may only be able to recognize it accurately when it's between scale-degrees 5 and 6 in major (as it is in that tune). They'll likely have some problems hearing—and definitely creating without prior reference—that same major second between scale-degrees 3 and ♯4.