They do often give credit to partnering organisations and agencies in their press releases. Here are some equestrian examples:
http://www.rspca.org.uk/servlet/Satellite?blobcol=urlblob&blobheader=application%2Fpdf&blobkey=id&blobtable=RSPCABlob&blobwhere=1236788766162&ssbinary=true gives credit to World Horse Rescue and Redwings
http://www.rspca.org.uk/media/pressreleases/details/-/article/PressFiveBannedFromKeepingAnimals24Jul13 gives credit to Blue Cross and World Horse Rescue
http://www.rspca.org.uk/media/pressreleases/details/-/article/PressSixMonthJailSentenceForCausingUnnecessarySufferingToHorses plugs a number of horse charities
If, when publishing articles, the press choose to not print those credits then that's hardly the RSPCA's fault - is it?
Moreover, it's not all about 'saving animals' - the RSPCA deal with the people who own the animals, far more so than the animals themselves. Other animal charities do not have the will or the means to prosecute individuals, to seek bans and prevent people who are not fit to own animals from owning them. Surely, in seeking to prevent the unfit (as determined by a court of law) from owning animals, they are benefiting animal welfare? Other charities also have the luxury of being able to say "we're full" and turn animals away - knowing that the RSPCA will step-in and deal with it one way or another. Dogs Trust, for example, "won't put a healthy dog to sleep" - but they also won't take-on Staffies in many places or any unhealthy dog or one with behavioural issues, who are simply turned away.