Conditional conundrum

Solution 1:

First, Second, Third, or Nth Conditional are terms that have been made up by certain teachers of English as a Foreign Language, most of whom are not native speakers of English. They are used only by some teachers (and, of course, their students, who have no alternative but to believe their teachers). They are not terms used by English grammarians, and they are completely unknown to native speakers educated in Anglophone schools.

Since, as you say, Anglophone schools teach native speakers diddly squat about English grammar, settling on proper terminology is not simple. Everybody wants to use the terms they're familiar with, even if they're not sure what they mean, nor how to distinguish the First from the Nth Conditional.

There are only two tenses in English: present and past. There is, in particular, no Conditional tense, of any number. There are, however, many, many (hundreds, even thousands) of syntactic constructions in English. To express opinions about causation and probability, English uses a number of such constructions, but there are far more than 4 (or whatever the canonical number of "conditional tenses" is sposta be; I've never seen a coherent description or count, personally).

As for the sentence you're asking about,

• I think threats of stoning if they don't succeed would be more effective.

it's perfectly fine, because there's no requirement for the if clause to be in the past tense,
though that's OK, too.

• I think threats of stoning if they didn't succeed would be more effective.

Generally the tense shift in if...then expressions is optional, not obligatory,
especially if you're dealing with a potential condition instead of a counterfactual one.

Solution 2:

This centres around the position in time of the action.

If you are focusing on 'threats' this is referring to the team's success in the future, in which case 'don't' is most appropriate.

I think threats if they don't succeed would be more effective.

If you are focusing on the act of 'stoning', this will be referring to the team's success in the past, in which case 'didn't' is most appropriate.

I think stoning if they didn't succeed would be more effective.

(Call it a 1-1 draw...)