Sounds like a great project! I'll be following with interest. I played 2E for years, and have good memories of the system. Here are my thoughts:
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* Keep the ability score tables as is, with the possible exception of removing exceptional strength (%). I personally abhor it, but it is a legacy item. If the overwhelming support is to keep % str, I'd keep it.
This is a keeper. Exceptional strength is simply a fighter class ability.
* Races: Human, Dwarf, Gnome, Elf, Half-elf, Halfling, Half-orc and Half-ogre. The latter two got pushed in Skills & Powers, and would make decent additions. Other races (orc, goblin, hobgoblin, minotaur, bugbear, ogre, kobold, pixie, and tiefling) could be inlcuded later in an additional supplement. Races would be mostly left the same, with an additional human benefit for removing level limits.
Looks OK. The level limits were so high as to be meaningless anyway. I'm all for giving humans a generic benefit to bring them in line with the other races. If you make it appealing enough, you'll end up with the humanocentric default setting that EGG wanted.
* Classes: Fighter, Ranger, Paladin, Cleric, Druid, Specialty Priest, Thief, Bard, Mage, Specialist. Dual-Classing would be eliminated (or regulated to a sidebar) and humans would be allowed demi-human style multi-classing (with restrictions). Wild-Mages, Assassins, and other 2e core-classes would have to come later.
Here's where a big fix is needed. Fighters and mages lost something pretty significant by multi-classing--the ability to specialize. Clerics and thieves, on the other hand, didn't lose anything. So some multi-class combinations were reasonably balanced, while others weren't.
In my mind, each class that can be part of a multi-class combo should have at least one significant power that is available only to single-classed PCs. I think this would go a long way toward fixing the issues with multi-classing.
For a generic setting, make clerics 'priests of a pantheon' and specialty priests 'priests of a god'. It shouldn't be too hard to come up with a construction kit that more or less mimics faiths and avatars. For example, if the god has a warrior aspect, the specialty priest gets broader weapon access, multiple attacks as a fighter, and a flavorful spell-like ability every odd level. If the god is more peaceful/magical, given them some "always on" powers and a capstone power at name level instead of the extra attacks.
And finally, surely we can come up with a better name than 'specialty priest'!
* Non-Weapon Profs would remain similar to original concept, as glorified ability checks. Right now, I'm not sure in what direction to take them, other than towards a true skill system (akin to 3e on). I feel they are too important to drop though.
I never had a problem with these, as long as the DM didn't prevent us from riding horses or starting fires even if we didn't have the proficiency. Each character should have a reasonable pool of general knowledge.
And devoting a second slot to the same proficiency should give a significant bonus.