I think Stardew Valley/Harvest Moon style "farming" mechanics are far too binary and feel like a factory rather than a farm or garden. I also think that watering is a poor choice of stand-in for all the repetitive farm maintenance tasks. So here are some specific suggestions for extending those mechanics in (relatively) easy ways to make the system feel more organic.
TL;DR version:
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Use temperature and crop grows-well/grows-poorly/dies temperature ranges instead of a binary "crops die when the season ends" mechanic.
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Use weeding (minigames?) instead of watering as the stand-in for repetitive farm tasks.
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Have a climate where plants often get enough water from rain, but you have to address flooding/drought/consistent dampness. Consecutive dry days slow growth, consecutive days with rain increase chance of mold/fungus diseases.
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Also, animals poop!
Instead of crops dying when the seasons change (yes/no), have the seasons control the temperature (and maybe day length). Each crop has three temperature ranges: grows well, grows poorly, dies. Throw in a little randomness on the temperature to make things more interesting. Bonus points for randomness on a bell-curve so you have a 5-year or 10-year rare late frost.
This gives you more interesting decisions: do you plant tomatoes and hope you don't get a late frost? Do you plant lettuce late spring and hope you don't get an early heat wave to make it bolt and get bitter? Plus it opens the door to mechanics like saving your crops from frost with floating row covers or cloches, smudge pots or misting; or using shade cloth and misting in the summer to protect against excessive heat.
This also addresses the problem of the scheduling being so static in Stardew Valley: now you still get a crop if things are a little off, it just takes longer. It's not just "plant this thing, 10 days later you get a crop, then every three days after that (or it dies if you screw up)." Bonus points if you get less produce or less valuable produce for poorer growing conditions.
Watering seems like a poor stand-in for all the plant maintenance tasks. Especially with it being binary (crops die if you don't water them every day) it removes the ability to have much dependence on the weather. A rainy day just saves you the task, there's no drought, flooding, mold from being too damp, etc.
I think I'd use weeding instead: zoom in for some sort of weeding mini-game? Choices of whether you pull weeds by hand (slow, accurate, can get right up to your plants), a hoe (faster, may not kill 100% of weeds, can't get right next to your plants or you may cut them off too), or tractor with cultivator (needs straight rows, space for the tractor wheels to go, etc.). If you're hand weeding, do you take the time to push the leaves aside (or zoom in to look under the leaves) to see if there are weeds there, or just pull the ones you can see?
Bonus mechanic: allow planting anywhere, instead of on a grid, and plants grow poorly if there is too much overlap between their "root zones". And straight rows make hoeing or tractor cultivation easier.
Then watering could be used to make you care about the weather: most farming you want to match your crops to your climate so you generally have enough water from rain, and you only have to deal with it if there's a long dry stretch. How do we make a very simple mechanic about that? What if crops grow slightly slower every day they don't get watered (and eventually die) but there's a chance of mold/fungus that goes up every consecutive day that they do get watered?
Bonus if you have drizzle versus heavy rain or rain part of the day. And you could have things like making hay where you need three sunny days to dry the hay or it's spoiled. But you could do silage or haylage instead, if you have the setup for it.
Also, where there are animals, there is poop! Both the negative aspects of this (oops, I stepped in a cow pie!) and the positive (hey, free fertilizer!) are important parts of the farming experience. :)
Bonus points if your pigs like to roll in the cow pies and your chickens like to kick them apart and eat the bugs that lay their eggs in them.