If the apostles had passed on tradition to the early church in addition to the scriptures, then it would be expected that what was additional would be in accord with the written documentation.
What Protestantism rejects is the concept that the apostles could have deliberately passed on traditions which are contrary - in both content and tenor - to what the gospels and epistles set forth. Or that they would have passed on what was supplementary to the degree that it was 'further revelation'.
Article Six of the Thirty Nine Articles of the Church of England states :
Holy Scripture containeth all things necessary to salvation: so that whatsoever is not read therein, nor may be proved thereby, is not to be required of any man, that it should be believed as an article of the Faith, or be thought requisite or necessary to salvation.
Importantly, this articles refers to 'whatsoever is not read therein nor may be proved thereby . . . is not required of any man.
Protestantism holds that one of the very reasons for the Divine Provision of an inspired and infallible documentation - the Greek scripture - is that historical traditions, even if, as the OP question states, those traditions are 'close to the source' are just not reliable enough.
The words of Jesus Christ himself, and the words that he gave to his apostles, are those which are documented in Holy Scripture. Protestantism, in its original form, holds that nothing else is to be considered above scripture in fact or in authority.
Faith cometh by hearing, and hearing . . . by the word of God. Romans 10:17.