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This first paragraph is simply untrue. You are always holding a valid signature to spend the latest funds.

The additional assumption vs simply holding your own funds is a throughput requirement: that miners not censor your transactions for some (up-front-chosen) period of time.

With normal funds there is a non-censoring requirement, but it's more vague since the miners may have to censor you forever to make your funds useless.

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It is not. If we have an open channel and I send a payment to you and confirm that you have received it (and perhaps that you sent some good my way), I can then close my channel on the BTC network claiming that no transaction happened. Lightning does allow you to punish this, but only if you find out in time - otherwise, the transaction is essentially rolled back - but the goods you sent are obviously not. This way, I can double-spend my BTC and roll back arbitrarily late into the future (assuming the channel stays open).

And related to censorship, I'm not sure what is the point you are trying to make. Still, censoring forever is not hard (you just add some wallet address in a list, and look those up before choosing which transactions to include in a block), and is completely equivalent to how payments censorship works in regular banks as well.

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