Is crit chance REALLY important enough that you couldn't just remove it? I can see how it adds a bit of nuance, but would the game really be a lot WORSE from removing it?

These are the main concerns about removing crit. They aren't deal-breakers, but are things we'd have to consider or fix.
1) Auto attack champions damage starts to feel really predictable. You know before a fight starts what the outcome is going to be.
2) Closely related, fights start to feel really flat. This is more of a feel issue while #1 is more of a skill issue.
3) There is one less stat on items for certain champs to care about.
4) We lose a hook for mechanics ("after a crit...")

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