I misspoke earlier. I strongly believe that sand is not a good amendment for clay soils. The particles are simply too small. As long as you add lots of organic matter with all of that sand you purchased, you might get away with it. Sand can work only if you actually remove clay and replace it with sand. Otherwise, those little particles will pack up with the clay particles. (Unless you add lots of OM, too.) Soil aggregation is created by the microorganisms. You aren't going to achieve it any other way. They will survive ONLY in a soil that has the organic content to support them. Beneficial fungi and bacteria are what a good soil consistency is all about. Yes, organic amendments eventually decompose and are turned into microbe and plant body, but the addition of a continual supply of OM is not difficult or challenging. In a planting bed, this may mean an annual application of organic mulch of some kind. BTW, some organic fibers take years and years to decompose, so continue to provide texture to the soil. On your lawn, it means that you let your clippings return to the soil when you mow. A healthy stand of turf returns a huge amount of OM to the soil just in the form of root sloughing (as mentioned earlier). Now, about that gypsum! If it were only true! Well, actually, the miracle IS true in some cases. If you live in an area where your soil is sodic (excess sodium), and the lack of soil aggregation is BECAUSE of that, gypsum will help. The chemical reaction of the CaSO4 takes the sodium out of commission, so to speak. But if your soil is just plain clayey....it won't do a thing, other than add calcium and sulphur to the soil. Thus, the gypsum miracle is not a total myth, lol! If you have naturally sodic soils, you'll find that annual applications of gypsum will help (along with everything else). I have brick hard clay soil. With very little initial improvements, I find that plants of all kinds thrive in it, as does the lawn. Here is a link that might be useful: See if this makes sense.....

When it comes time to put the Pavers down. you will want a 6 to 8" base of compacted Medium gravel. Medium gravel allows the pavers to lay more flat/level and allows good drainage. Crush & Run is a little to coarse with so much rock dust in that sets up like concrete and does not drain the best. And you do not want to put Landscape fabric under it. Landscape fabric is fine and highly recommended if you wish to just place the gravel on top of it like with Emma's egg stone or mulch for it helps with weed control. One method we use here to keep animals from digging is to put down chicken wire then the type of stone we will use to lay pavers on if the project is to be delayed or held off (such as in your case) until a later date. The chicken wire being thin will easily compact into the dirt with the gravel and not effect compaction. If an animal does dig such as a dog. It will stop once it reaches the wire.

The yard does not have to look like a recycle salvage- the cheap/recycled pots can be camouflaged with the much nicer wood planters just like garden nursery pots. In the case of using the square storage bins or kitty litter buckets, they fit/fill a bit nicer into a square wooden box surround than round pots can. Garbage cans don't have to look garbage either- there are attractive ones out there. Or at least ones that more decorative or meant to be seen and look like pots and don't necessarily need to be hidden. They can also be handy for container growing stuff with deep roots. I container grow horseradish and salsify, and just couldn't find any pots that were big and deep enough till I found a nice trio of plain round heavy plastic garbage cans- they look a lot like the big pot on the left in the OP pic, only they came in black.

If my guess is right, then I think scraping out the "grout" between the stones would only allow water to accumulate in the remaining space (assuming it can't drain freely through the compacted stone dust beneath) which may actually increase the wicking problem. I would begin by doubling back with your contractor on this, to find out what they did and why they did it that way. My concern is that even with better drainage beneath the stones (e.g. coarse sand), using stone dust as grout neither seals the edges of the bluestone to repel water off the surface (as grout between bathroom tiles does) nor allows water to drain through freely. Unfortunately, the best solution would be to pour a concrete pad, mortar the bluestone to the pad, then use true grout between the stones. Otherwise, if you do keep an aggregate base (e.g. sand), you would want the stones to be as tight together as possible, with nothing between them that would hold any water.