You bring up a good point. It may simply mean that in order to attain salvation, one has to do so using the teachings of Jesus (peace be upon him). I'm not sure that means to worship him.
All the prophets (including Noah, Abraham, Jacob, David, Moses, etc.) that came before Jesus called people to the worship of the One and Only God, Creator and Sustainer of the Universe. Part of this teaching says that one does not need an intermediary between themselves and God. This means that interpreting the word 'through' as an intermediary means that the bible contradicts itself.
That being said, if 'through' means following the teachings of Jesus, this indicates that the message of Jesus was not going to be the final message to humanity. He did say afterall to follow the "comforter, spirit of truth", who will come with the everlasting guidance for all.
Well, how do we know this 'comforter'? Jesus tells us that this 'comforter' will acknowledge Jesus and will deliver an everlasting message. Historically, the only religion that came after christianity and acknowledges Jesus is Islam. So the question remains, did he deliver an everlasting message? To date, the Quran is the only book that is still maintained in its original format, original language, letter for letter.
If the above analysis is correct, then 'through' does not mean "an intermediary"; instead, it means "by following the teachings of". If this is the case, then in order to comply with the teachings of Jesus (peace be upon him), one must at least consider Islam as the message of the "comforter" that Jesus has spoken of.
And God only knows best.
(Read between the blades)