That would require me to be able to talk... hard to do anything like that when one speaks with her hands ^.^
Monks use Ki, which is not actually an extension of the weave but personal spiritual power; the Monk's use of Ki is not magical, and it continues to be useful even in places of dead magic. The four-elements path is a... mechanically flawed method of communicating how those specific monks utilise their ki, but at the time of designing it, a better way that still fit in with 5e's simplicity philosophy was not found. It trades intention for simplicity; four-element monks aren't really supposed to have 'Spells', because they aren't weave-dependant magic, or they're not meant to be. They're meant to be expressions of Ki, like everything else in the Monk's arsenal, but it just didn't work out that way at the time. I've since seen at least one very good re-write of the four-elements monk (Way of the Elements - Revised, which I consulted on and assisted with editing and style language for, in fact, so, maybe some bias ^.^) that captures the spirit of the original intention very well - granting them abilities that have parallel effects to spells, but without just giving them spells... this method would have been considered too divergent from what could be achieved more simply with already established material, at the time that 5e was first published, and I feel to this day that that was genuinely a mistake.
What we have, unfortunately, if you don't use a homebrew re-write, is a monk who casts spells fuelled by Ki points - but because they're spells they can still be counterspelled or fail in antimagic zones and so on, which is not meant to be the case, the way it was originally intended... and honestly, the spell access that we ended up with ended up being too limited to really feel like an integral part of the class; you just end up being a monk with a handful of lacklustre spells that take away from your core resource - which is still better spent on your core monk capabilities - and foregoing any of the nicer perks that come with other traditions.
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Oh, and I forgot, regarding magic users:
Artificers are the newest kids on the block in 5e, and they also have a form of spellcasting. Artificers are much in the same situation as wizards, having no innate connection to the weave and no intermediary. they support their method with a dash of alchemy and a dash of mechanism and other little tricks to help serve as foci and reagents for actualising magical effects. Where a wizard relies on pure intellect and precision of movement, word and gesture, and artificer puts some of that work in the hands of formulae and construction (after all, a precise silver lattice that will hold a current of electricity is far more precise than moving your fingers over the same pattern and vocalising the right harmonic.... isn't it? Er... shaky hands are still a bad thing, regardless).
That said... It is
This alchemist's personal opinion that the Artificer as it was released is exceptionally lacklustre and a pale shadow of what it should and could be... and she highly encourages people to instead take a look at a far more comprehensive, flexible and versatile alchemist class... a far more
Definitive Alchemist, if you will... complete with six subclasses of varied and unique flavours, perfect for creating the exact type of alchemist you imagine ^.^ It's free! *cough* /ShamelessSelfPlug
