Broadly, my earlier approach was to focus on the single strong piece of evidence in favor of A, and ignore the weak pieces of evidence against A, whereas now my approach is to give some weight to the several weak pieces evidence, and to allow them it to overwhelm the single strong piece of evidence in favor of A in some situations.

That seems very commonsensical and should be very uncontroversial.

Well, it's hard to be controversial when saying "you should give some weight even to weak evidence". As opposed to evidence in favor of a position which would not shift your prior at all? That would be a contradictio in terminis, since evidence in favor is defined as shifting your prior to some degree). "In favor, even weakly" = P(A|evidence) > P(A).

What about the second part, which -- because of the "in some situations" qualifier -- is equivalent to "there exist situations such that a number of small belief shifts can overcome one large belief shift. Common sense as well, since it just restates that adding enough (non-diminishing and ergo terms of a diverging series) epsilons will overcome any fixed number, however large.

As a general rule, we should not expect there to be strong pieces of evidence in favor of a false position. Not strong in the sense of "well presented", or "given by an authority figure", but strong as evaluated by a person knowledgeable in the field. In physics, you won't get to 5 sigma using many small updates, you'll have a couple of strong results. There may be the occasional strange circumstance in which there appears to be actual strong evidence against a position which (the position) later turns out to be correct. However, there always will be a lot of weak evidence (at various values of 'weak') for or against anything, it's just too easy to come up with using motivated cognition.

So we should take note if there's strong evidence involved in any issue, but with "you should not outright ignore weak evidence", we can all be friends.