The problem is there's not a single club usable by a non-elf druid that is +1 enchantment in EA, and staves are bad.
My other problem with your suggestion of making staves versatile is that it only exacerbates the problem of making shields mandatory for anyone but great weapon or DW martial classes, and quite frankly holding a two handed staff with a shield just looks so dumb. It's not the iconic look of a caster. Which is why the benefits of a staff should go toward being a desirable weapon for classes that want to cast. There should be viable, competitive alternatives to using a shield. I certainly don't enjoy kitting all my casters in medium armor proficiency with shields because 19-21 AC over 14-15 AC with mage armor and a staff that provides no benefit is too good to pass up. The only time you consider otherwise is going Warlock initiate on Wizard to try to get slightly more damage with Ray of Fire+Hex than Magic Missiles Rank 2, but at the cost of being far more unreliable in accuracy (imo not worth trading off 4+ AC).
Granted, the latter argument is more of a commentary on how awfully implemented casters are in this game, and how the Concentration mechanic on top of the limited spell slots is crippling their gameplay. Someone has to be on mushrooms to believe that spamming Eldritch Blast as a warlock because you got 2 spell slots and one is reserved for Hex happens to make for an engaging gameplay experience.
So it sounds more like a weapon itemization issue than a need to rework Shillelagh. Instead of changing the bounded accuracy system, I think it would it would be better to advocate for better itemized clubs and staves, which still might be added either in EA or later down the line.
And considering aesthetic is so subjective (I disagree with your opinion here, I've seen one handed stave/shield look good depending on the appearance of each), I don't think this is a reason to change the RAW 5e rules of spears and staves as versatile weapons. I generally prefer more options here where it makes sense, rather than less. Especially because there already is going to be a reason why casters don't use staves + shields, in that many casters aren't proficient with shields (wizard/warlock/sorcerer/lore bard). And if staves were versatile you could dual wield them with another item (weapon or otherwise), if that second item had a magical attribute more desirable than a shield, so Druids aren't any more stuck with a shield than any other class. Again, this is more an issue with magical weapon design, since Larian is making their own rather than using established items from the DMG. I don't mind them making their own, it's just harder because they haven't been play tested or designed with the ruleset in mind.
Also, staves aren't always caster items in D&D, Monks often use them one-handed, so they can make unarmed strikes with their offhand. So I don't think limiting staves to two-hands for "caster feel" fully takes into account all the ways in which staves are going to be used.
The good news is that as the player progresses past lvl 4, casters have more and more spell slots, and more powerful spells, so caster gameplay seems artificially weaker now with the lvl cap. It also hurts casters that they are getting spell damage according to the PHB spells, but martial classes are given a whole bunch of items that make their weapon attacks deal way more damage than they would in TT. Larianisms are more beneficial to martial classes (no class requirements on spell scrolls needs to be fixed before launch, the +3 weapon oil is super strong, potion of giant's strength + extra weapon attack from speed potion, easy advantage from backstab to negate lowered accuracy from great-weapon fighting, weapon dipping), so playing a caster like a caster feels weaker in comparison, especially before the more powerful high level spells can be accessed. Even warlocks, which are meant to be a more simple spell-casting class, could become a lot more fun once more eldritch invocation options are added (and obviously everyone is hoping for the hexblade subclass, but that probably won't be added before launch).
And while I don't see anything wrong with making a "caster form melee druid" in theory, it will be harder to keep that approach effective at higher lvls, once martial classes get extra attacks, and the druid is stuck with 1 main-hand 1 off-hand attack. I assume the reason Shillelagh doesn't affect both weapons is to reserve the value of the two weapon fighting style for Rangers/Fighters (otherwise no attribute modifier on offhand damage), although I'm ok with your change seeing as the druid isn't going to keep up with either of those classes in martial damage anyway. I say play what you want, but I also don't think the game should make every class equally good at every role. "Balance" in a team based RPG is more about making sure each class has their own niche to excel at, rather than making sure that each class always feel as strong at each aspect of the game, at least this is how D&D is designed (if they didn't agree with this, why make a D&D game).
I also saw you mentioned Clerics above, and I do think that part of the reason they feel so weak is that their most important offensive spells haven't been put into the game. The ability to do aoe damage with spirit guardians, +bonus action damage with spiritual weapon + regular weapon damage, would really add up, and can last for an entire combat using just two spell slots. Right now, we're just stuck with guiding bolt and inflict wounds. So I'm hoping most of these problems are just resulting from the fact that they haven't caught up with all the spells and abilities for casters in the PHB, because so far most of the problems are self-made by changing the game design from the TT rules, and then failing to rebalance properly.
