Do transactions need to be accurately timestamped?
No, because Bitcoin is not about measuring time. It is about double-spending problem.
Which means, if you run for example regtest, then you can include a lot of mainnet transactions directly, even though they happened in a completely different network, on a completely different time. If regtest would have 21 million coins, it could be copy-pasted "as is", but because amounts are different, it is somewhat limited, because you cannot copy-paste for example a transaction with 500k BTC.
So yes, you can take some mainnet transaction from 2009, and include it into regtest, in 2023. Which means, the time is not really enforced, because if you can clone the coinbase transaction, and if you can clone the whole flow, then you can perform 1:1 testing, on exactly the same transaction hashes. But as I said, there are limitations, like coin amounts, that prevents users from using regtest as a mainnet mirror (but probably testnet3 or signet could still do that; and it is a proof, that time is not really enforced, and transactions are reorg-resistant).