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Maybe there is a reason this would not have worked, but I don’t see it.
There were options involving the debt ceiling too apparently.
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That article concludes by saying "With a Republican majority in the Senate, Democrats have had no recourse for stopping or delaying Barrett’s confirmation." And ultimately, there wasn't. The Republicans controlled the Senate, so they controlled its procedure. If the Democrats had moved to impeach Barr, the Republican majority could and would have moved to immediately dismiss the impeachment. Might not have looked good, but given what was at stake, I have no doubt that's exactly what they would have done.
Not without a majority in the Senate they couldn't.
True, for situations where similar means completely different! When Obama was trying to nominate a new SC judge, the Republicans had a Senate majority, which they had held since 2014. That makes a significant difference.