I went to school in the UK in the 80s where we were taught ZERO grammar.
I had to teach myself before becoming an EFL teacher.
We teach the conditionals to adult EFL students in the following way:
general truths: zero conditional
present, present - when it rains, the ground is wet
future possibility: first conditional -
if +present, future - if it rains, you will get wet
hypothetical / unlikely future: second conditional
If + past , would - if it rained, you would get wet
past regret: third conditional
if + past perfect, would have - if it had rained, you would have got wet.
and mixed conditionals -which are far more common in natural native speaker speech. (e.g. if you had called me, I wouldn't be mad)
It's really tricky for EFL learners and needs lots of practice/exposure.
My partner is a primary teacher and tells me the conditionals are taught as subordinate if clauses from Y2.
They are not taught in the same way as to adult EFL students as primary children (should) already have a firm understanding of the concepts as they are (should be) proficient speakers. They need help to identify them so that they can use them in writing.
The common native speaker mistake of writing 'could of' etc is due to a lack of awareness of the features we use in our connected speech - we have a stress timed language where we change the stress on words/sounds depending on the context, stressing the important information.
'have' is both a 'real' verb ( I have a cat - never pronounced as 'uf') and a 'dummy/helper/auxiliary' verb (I would have helped - always contracted to a uf / of as it doesn't carry any important information.)