Which is to say, one party is certainly responsible for picking this fight, but both parties are responsible for not ending it.
They are acting like children. If your son relentlessly bullied your other baby and in order for your baby to stop crying your son said "give me $5" (a meaningless amount to most adults) - the solution wouldn't be to give him $5 and teach him that whenever he wants money he just has to threaten throw the baby down the stairs.
If the clean CR fails, then its a terrible future ahead. In 2018, when Democrats were in the opposite situation (they were denying the CR), it was on the onus of the Republicans to push for the CR.
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The only reason why the CR is agreed upon in the past, is because both sides recognize the harm that a shutdown would cause. Unfortunately, one side is willing to shutdown the government and avoid the CR this time.
I don't think the democrats should cave because this is a terrible precedent that far outstrips "appearance". It feels like a perversion of checks and balances where, because on side doesn't get what they want, they shut down the entire government until the other side caves. I won't be a fan of future presidents shutting down the government every time they can't come to a bipartisan agreement.