If I remembered one day, I would choose to remember my wedding day.

(Grammar Questions I Couldn't Answer)

A co-worker of mine was preparing a lesson on unreal conditionals (i.e. 2nd conditionals).  

For the presentation stage of the lesson, he wanted to build a context around the sentence "If I could remember one day, I would choose to remember my wedding day."  (I believe the intended context was that if he could choose to preserve one day of his life perfectly in his memory, it would be his wedding.)

However, in the course of preparing the lesson, he realized that the textbook (Reflect 4 Listening and Speaking, Unit 4, Grammar p.62: Unreal Conditionals) defined the unreal conditional as:

If + past simple, would + bare infinitive.

He was worried that his example sentence didn't fit the pattern because he was using "If I could remember"--i.e. If + could + bare infinitive."

This sparked a debate among the staffroom as to whether "could+bare infinitive" was the same as the past simple in this context.  I took the position that it was.  (Certainly that's the way I've been teaching the 2nd conditional for years.  And I showed my co-worker worksheets that I had in my archive like this one in which I had used "could fly" as equivalent to the past simple).

But there was also a secondary question: why do we even need "could" in this sentence?  Why can't we just say: "If I remembered one day, I would choose to remember my wedding day" ?

And what was the difference between "If I remembered..." and "If I could remember..."

The whole thing reminded me of a similar grammar question I couldn't answer 4 years ago about the difference between "I wish I played with that dog" and "I wish I could play with that dog".  I wrote about those sentences in this blog post here.  I suspect they are related cases.