a true and faithful doctrine does not lead to people being burned alive. thats not the way of Jesus.
That people were burned alive for rejecting the doctrine of the Trinity, does not, in itself, make the doctrine of the Trinity untrue.
It just means that certain adherents of the doctrine were not born again of the Holy Spirit.
This does not mean that one must not adhere to the doctrine in order to be born again of the Holy Spirit.
As with the red dog analogy, there are exceptions to every rule.
If you must believe that the doctrine of the Trinity is true in order to be saved, this fact is not necessarily disproven by the fact that some who hold that doctrine were not saved. For the Trinity is not the only qualification. One must receive Jesus as their personal Lord and Saviour.
Some, who are religious, may have sound doctrine in their minds, and be zealous for that doctrine, and yet not be born again.
Because faith is not only a matter of the mind but of the heart.