I thought of that as well, but how would the client know if a previously obtained list of transactions had been invalidated?
Huh? I don't see how passing <txid> makes that problem any better.
Let's say you've got transactions A B C D (in that order). With times T1, T1, T2 and T2 (A&B happened 2 blocks back, you got C&D in the last block).
Now imagine a block-chain re-org, so B and C are now invalid.
Ask for all transactions after D and you get nothing-- same as if you asked for all transactions with time greater than T2.