I wish I played with that dog

(Grammar Questions I Couldn't Answer)

This question actually comes from a colleague, who asked me about it in the staff room.

My colleague had been teaching the grammar point "I wish" out of the textbook English World 6 Unit 10 p.106.
English World 6 presents the grammar the same way all the other textbooks do: "I wish + past tense", for example "I wish I had more money."

My colleague was illustrating this for his students.  "So 'I want to play the guitar' becomes 'I wish I played the guitar'.  And 'I want to play with that dog' becomes 'I wish I played with that dog' ."

However, at this point, my colleague's native speaker intuition immediately told him that "I wish I played with that dog" was not correct, and that the correct sentence was "I wish I could play with that dog."  However my colleague could not explain to his students why the addition of "could" was necessary in the second sentence, but not in the first.  Nor could he explain why this sentence needed a "could" when the textbook simply gave the formula of "I wish + past tense".

So to summarize:
"I wish I played guitar"--sounds grammatical
"I wish I played with that dog"--sounds ungrammatical

"I'll be honest," I said.  "I've never thought about this until just now.  But maybe we can puzzle it out.  I wonder if "play guitar" is describing a state of being able to play guitar, and is thus more of a state verb.  Maybe state verbs take the past simple, but action verbs need "could" ?

This seemed like it must be a fairly common usage, and I expected to find this explained in a grammar reference book.  But we consulted several grammar reference books on this (including Practical English Usage by Michael Swan and The Grammar Book: An ESL/EFL Teacher's Course by Marianne Celce-Murcia and Diane Larsen-Freeman) and could find nothing about state verbs or exceptions to the "I wish + past tense" formula.

Having thought about it more, I wonder if the distinction is not between state verbs and action verbs, but between habits and present actions.  So, "I wish I played with the dog" would be acceptable if you were describing a habit--e.g. "I wish I played with that dog every day", whereas for a desired present action, you would use the past continuous--e.g. "I wish I was playing with that dog".

That's my best guess.  Anyone else have any thoughts?