I think that some edge cost repricing/rebalancing is required, but is there really any glaringly underpowered class or race for which incorporating any of the generally picked "must-have" edges would be justified? The "must-haves" tend to be edges that make an already good and useful ability even better, so incorporating them for everyone seems not only unnecessary but probably unbalancing too. However, I guess there are a number of minor edges which could likely be factored in without creating further problems (Sacred Word comes to mind).

Many edges have been added for key abilities to improve them, for abilities that are going to be used anyway, edge or not (e.g. Steal Sight & Psyche Crush). I think this is perhaps the major reason why certain edges are picked by everyone eligible. Also now with less edge points available the choices gravitate even more towards the "mandatory" ones. People of course want to choose the edges that will generally be the most helpful to their character (though e.g. their enemy range might cause some variation and have them pick another instead of the "obvious" one).

IMHO the focus on edges should rather be improving the various less-used/niche skills and/or race/class combos, making them more viable options (e.g. the elf/delf/svirf special shifter forms) and thus creating more diverse edge choices among the players.

All that said, I think you're kind of focusing on the wrong thing here. It seems that according to general opinion the issue with edges is edge point acquisition (i.e. the repetitive exp/obs grind, but I won't rehash that here). People seem happy with edges themselves and the choices available, even if currently the choices may be very concentrated. Also to have more diversity with the more popular/powerful edges, I think rather than trying to incorporate them into existing skills/classes/races, a more viable course would be to create alternatives of similar level, so that there's a real choice (due to functionality and cost; also some down-tweaking seems called for).