as far as i know SegWit is a soft fork and it is backward compatible. in other words all the old keys will continue to be valid and they will continue to work.
but it is best to upgrade though. even without SegWit you should always try to keep your wallets upgraded, sometimes they find bugs and fix them which are critical. and if you can't find that wallet then just install another one and send the balance to the newly generated keys.
I made a test transaction and it went through. The transaction is shown in the wallet as well as online via a block explorer. So it seems to work. *whew*
And yes, you are fully right with respect to updates. However, the thing is, the wallet is not updated so often. And now, it seems it is not available anymore.