What effective methods exist for visual learners to get Chinese grammar?
Full Disclosure: I haven't tried this for studying Chinese but in English it seems to work quite well. I haven't a Doctorate in Educational Sciences or the like but have been teaching grammar to all ages and many nationalities of close to 10 years.
The method is based on Cuisenaire rods(coloured rods of differing lengths, the smallest being 1cm long(white) and the longest being 10cm(orange) the intermediate rods increasing in length one cm at a time.) While originally these rods were used to explain complicated mathematical concepts visually, Caleb Gattegno brought this concept initially to teaching literacy, and then to learning foreign languages. It came to be part of a teaching system called the Silent Way(since then has largely fallen out of favour.)
OK, done with history. Gattegno used these rods to replace parts of speech(or syllables for sentence stress) and then would drill the students on proper placement. With my students, and a handful of different colour pens, we would construct sentences based on colour. The learning curve was steeper than many educationalists had claimed and I finally found it worked for extremely visual learners, and those with ADD(in my experience two boys 13 and 15 years old).
As an example take the sentence:

我想买几个方形的扣子

Since we already would have established the parts of speech as having corresponding colours, for the sake of the example.

我 is blue
想买 is green
几个 is orange
方形 is orange
扣子 is red

So when you see this sentence your mind should automatically associate the part of speech/colour with that position and by imagining other words of the same colour inhabiting those regions reinforces the structure(drills).
Perhaps these structures will help to the explanation:

他把他的报纸给我
他给我他的报纸

把has no colour as a participle.(In English, I left the prepositions, articles and conjunctions, etc. black(vid. no special colour)) Because I am familiar with these colours I can easily see the grammatical difference. My tutor can explain the difference semantically.
I don't believe I can add colour here and I haven't found an organization online that is using the same methodology. I am currently training my Chinese tutor in this method because I find I know it very well and it works for me. Also, I have helped many adults pass government exams on grammar and writing using this method. As you said, I feel textbooks hinder rather than help and have slowly been building my own &quot;textbook&quot; according to this and other methods I like for myself.