This note is about Barbirolli's account of the Mahler Sixth only. I'll leave the Strauss for another day, perhaps.
But, regarding the Sixth, I had the rare good fortune to see Barbirolli conduct Mahler live (Symphony No. 9) shortly before his death. It was a sublime experience, actually better than his estimable recording of the same work.
His Sixth is, in my opinion, one of the great recorded musical statements of the 20th Century. I have lived with the recording in question here for some four decades. It is not the only great recording of Mahler's Symphony. However, there is absolutely no question in my mind that it belongs in every music lover's library, and should be basic recorded repertoire for Mahlerians everywhere.
It is heart-wrenchingly magnificent.