The Church does not teach salvation of works only.
The Baptism and the Eucharist theology you describe is not scriptural, except that, indeed, when someone is baptized either he has to have made an informed decision about it or if he is not of age or not of sound mind, then he needs a sponsor who decides for him.
A sponsor? Nobody can make a decision like that for anyone else.
Do you believe you have to be baptized to be saved?