But the Independent and the Sunday Times aren't really gutter press are they? I forget whether Harry has/is suing either of these publications, so can't comment on whether they might be acting in retaliation.
I think it's possible to watch Harry and Meghan's Netflix documentary and read Spare, and still believe that these articles might have something to do with Harry and/or someone close to him.
Harry would have a vested interest in having access to royal properties: it keeps the royal connections alive for an American audience; it gives credence to the idea that he could be called upon for COS responsibilities, no matter how unlikely that may appear to the rest of us; it provides free (to him) accommodation which might otherwise have to be paid for; it provides additional security, which is clearly important to him. It may also provide the way for the healing of damaged family relationships, which may be a motivation for him.
With all of these reasons for Harry to be interested in staying in royal properties, it is logical to think that he may put out 'feelers' via the press, especially if communication has broken down completely with the RF (as seems to be the case, particularly between him and William). Harry may not have many avenues to make his wishes known.
Unfortunately for Harry, it seems that the Palace have shut down his requests pretty quickly. (Although not brutally - he is still welcome to visit/stay, just with some notice).