Here's where we differ in interpretting narrative stuff from the game mechanics...I would posit that "doing miracles" is what any/all casting is supposed to be. That magic, of any type, functions/works is a spectacular thing. You don't have to legit be raising people from the dead or conjuring elementals (that's fairly high magic, still, right? 5th level? 6th?) for magic to be "impressive."
To the "bards are/should be full casters because they're supposed to be jacks of all trades. So they need to be 'good' at everything." Being "good" at everything and having full access to all levels of casting are not the same thing. The "jack of all trades" thing, by its very definition, means you are "good" at doing a bunch of different things. You are NOT, by definition, "the best" at anything.
I would submit that "full casting" is being "the best" at [your particular sort of] magic. Seeing as the game doesn't see fit to actually give the Bard its own kind of magic, they're just another "arcane caster," they have no business with full access to all spell levels.
To use an example/another angle, I can safely say -without any hint of arrogance- I am a pretty darn good cook. I can make a phenomenal dinner for two, multi-course dinner party for 4-6-10, or even (not often, but a few times a year) whip up catering for large gatherings/events/holidays. My repertoire is broad. I have several "signature dishes" friends expect and enjoy when they come over -for everything from an informal cocktail hour to Thanksgiving/Xmas feast. My collection of cookbooks is diverse. I can say with all confidence, I am a "good cook."
That doesn't mean you could through me in a restaurant's kitchen and say, "Go to town! You know all of this. Just make everything for this restaurant's intercontinental menu, any number of people that come in, any of day of the week." I have not been to -let alone successfully graduated from- the Culinary Institute of America (or France or anywhere else). I am not certified nor have the experience to be considered -nor would I necessarily be able to step into the role of- "chef."
The Bard can be a perfectly "good cook," perhaps even "above average" if they focus their diversity of talents into that area. They are also an adequate gardener, have a passing knowledge of first aid, history, mythologies, and general wildlife/animal idenitfication/behaviors, etc...
The Bard is not, nor should be expected to be (if we are using/keeping with a "jack of all trades" design sensibility), the chef, professional farmer or botanist, EMT or surgeon, professor or park ranger/zoologist.
Bard can certainly "lean into" any of those areas (subclasses), but solely at the expense of some expertise in one of the others. And, even then, they'd still be better than Joe Schmoe from da Block.
Long story, long. I do not think "they're a jack of all trades" can justifiably be used as an explanation for the "full caster" decision. But, that's just me.