https://en.m.wiktionary.org/wiki/Reconstruction:Proto-Germanic/gernaz would seem to indicate yes. Cool!
They would gladly sing together. Like is not needed because you already have gladly, which implies the like (I don't suppose you would often be glad to do something you don't like).
I remember gjerne as 'gladly' too, and 'vil gjerne' reminds me of 'yearning' which, whilst not an accurate translation, helps me remember the meaning of the words.
When I looked up "gjerne" in an online dictionary, one of the possible translations was indeed "willingly". But as far as I understand the english meaning, "willingly" emphasizes that people WEREN'T FORCED to do something. In comparison to that, "gjerne" emphasizes that people WANT to do something because they LIKE doing it.
I would expect "willingly" as a translation for "gjerne" in a context, where people were more or less forced (e.g. blackmailed) to do something for someone else and when someone thanks them the person forcing them replies with something like "Oh, the loved to do it." insert evil grin.