All very fair and well-informed comment there, CWS ; and it so happens that I’m very much a Martin Lewis disciple on such matters, meaning that if necessary I would always be inclined to handle my own claim rather than ‘farming it out’.
Nonetheless, these ‘specialist’ claim companies who, as you say, will often cherry pick (surprise, surprise .... it was ever thus where compensation matters are concerned) would never have found themselves a (viable, long-term) market in the first place IF the airlines were more pro-active in terms of both raising awareness and then meeting their legal responsibilities in a more ethical, and professional manner.
Furthermore (and I’m certainly not the first to raise this), I do believe there could be some merit in the introduction of punitive payments, ie. over & above the currently prescribed levels, to be made by carriers who wilfully and erroneously attempt to deny legitimate claims, by whatever means. I’m not saying it would be a total solution ; but it would sure help focus the beancounters’ minds somewhat.