If I were to make a Shaman Class, how would I do it...
Well. First things first they'd need to be casters. Primal casters, specifically. But would they be full casters...?
The Warlock Chassis could be a really interesting way to differentiate the Shaman and Druid class while retaining "Full Caster" ability and also free up a whole bunch of design space across the class, what with room for Pact Boons and Invocations on the Warlock shifted to something new for the Shaman...
They'll absolutely need to be Ritual Casters. Whether it's true Tungus Shaman, Native American Medicine Men, or anyone in the vast regions between tribal religious and mystical authorities almost invariably perform or head up the larger scale (I.E. not personal or private) rituals on behalf of a group.
Shaman are often depicted using only simple weapons. Ceremonial Daggers, Spears, Staves, but also the occasional hand-axe. So simple weapon proficiency makes good sense. Though maybe even make it a taboo for them to use more complex weapons. Not "Metal" like a Druid, but Martial. It's not that they -can't- gain proficiency, but maybe breaking taboos means you can't advance as a Shaman 'til you perform a cleansing ritual? Still "Level Up" your proficiency and HD but gain no class abilities.
Subclasses to cover the broad-strokes of Shamanistic beliefs. Spirit-Speakers, Medicine Men, Runecasters, and yeah at least one Shapeshifter. But to get away from "They're just Druids" we could keep their shapeshifting on a more "Human" grade, where they're acting more like Lycanthropes than beasts. Half-Human Half-Jaguar. Or a human-like Crow. Could tap into Barbarian a bit and assign them Totems with similar shapechanging functions.
Invocations would work just fine in the design space, even with the same name. Just change the availability to get away from Fey/Demons/Old Gods and more towards Nature, Spirits, and specifically Elemental interaction. Like calling up tiny elemental spirits to do various stunts. Air Elementals that lift you into the air, Earth elementals that grant you extra durability or fortitude. Things like that.
And then reclassify "Pact Boons" as something closer to "Shamanistic Tools". So you could have Instrument Shaman using music to provide bonuses to allies or debuff enemies, Headdress Shaman who dance and cry out to intimidate enemies or direct allies like a 4e Warlord, and Warrior Shaman who get to improve Spears and Daggers in some way or another and take to melee, themselves.
Do it in Level Up rather than standard 5e and you get even more design space since the Invocations are broken out a bit and players gain social/exploration stuff outright.
Could be fun.
I would like to note, however, this important aspect: While there are stories of Shaman being shapechangers throughout various cultures they are typically -antagonistic- in that aspect. Or, at least, untrustworthy. It's only really in the Malay and Polynesian traditions that transforming into a full-on animal is considered an outright positive. Many other cultures use shapeshifting shaman as a cautionary tale.