In which cases is it not true? Perhaps some examples might be useful?
If I had an AONE* or a DONE*, I cannot believe that any OW airline would refuse to give me a seat in coach on an all-coach flight (that was not oversold and had available inventory in any class). Are you saying that there are situations in which a OW airline would refuse to accept an A or D coupon for an available seat in coach?
As listed by Kiwi Flyer, there are a few specific cases where a DONE/AONE will book into Y , but they are the exception rather than the rule. Generally , the only valid classes are A,D and L
If you have a DONE and want to rebook into economy, in general you will need L availability and you should be prepared for the surprise when they will not rebook you if there is no L
Dave