Actually, you can get long sword proficiency, for example by being an elf. Right now armor proficiency is only done by category, but nothing prevents them from giving out proficiency in individual types of armor. That was kind of my point about tools -- they are very specific and visible things that you can be proficient in, just like individual weapons.This is not entirely true. You do not learn long sword proficiency or chain mail proficiency. You learn Heavy Armor proficiency a whole class of armors and you learn martial weapon proficiency again a large class of items.
I do like your your "weapon groups as skills" proficiencies idea. That's how a lot of other RPGs work and it makes a lot of sense. For D&D, you could also go the other way and make skills more granular and specific, more in line with tools. This becomes a problem keeping track of what skill you might be using since there's usually no visible object to remind you. Also some skills would be way better than others (Perception, for example).
I really like the Microlite20 skills; there are only 4 skills (Athletics, Communication, Knowledge, and Subterfuge) but they can be used with any ability. For 5e I'd probably alter the grouping slightly, maybe Athletics, Influence, Lore, Perception, Subterfuge, Survival.