Does `importprivkey` need a rescan even in a txindex=1 node? Documentation for importprivkey RPC command has an optional boolean argument called "rescan" which hints that the user may want to rescan the blockchain to be able to see previous deposits to this address.

Is the rescan needed if the node is configured as txindex=1?
Yes. txindex=1 is not related to the wallet and the wallet does not know about the txindex nor does it care. What the wallet really cares about is transaction output information and which transactions are related to addresses stored in the wallet. The txindex does not store any of that information, all it stores is the location of each transaction identified by transaction id. The actual contents of that transaction are not kept except in the raw transaction itself.