Honestly I think the drive looks pretty good. Transverse e.j.'s would have prevented any side to side movement, imho.
This is going to be long again. Fixing these types of things is difficult to put into words, but fairly easy once the basic concepts are understood.
The piers will likely do little to resist over time. The forces moving the slab sideways will almost certainly push the pier over, or plow it through the soil. If a long deep curb had been put in place of the piers, ir would likely still move. It almost certainly is the expansion contraction.
I sketched up a little drawing in iso view. Think of the cleat as a transverse curb, or anchor pushing back against the slab forced as the concrete expands in the summer heat.
Your pics include on of the pier where a sealed crack in the drive slab adjacent. That is a good indication the slab is moving toward the pier. and the crack appears as that edge stretches outward.
In the upper left is a representation of the rebar layout.
If mine I'd put transverse bars at 6" apart stacked vertically and about 8' apart vertically.
In the lower right is a cross section of a slab with correctly placed sealant. The top is the sealant in a hot day in the summer, the bottom on a cold day in the winter.
Ideally the sealant hould be placed in the mid temperature range.
It appears the joint was routed before the sealant was placed? Just a guess. It's very good to rout (with a special router) any cracks to be sealed. 1, it removes old sealant, 2, it cleans dirt and other material from the surfaces, and it eats and drys the concrete. Alternative methods can work carefully scraping to the depth to be sealed, and using a wire brush on a gas powered lawn edger, etc.
If the wrong sealant is used it won't properly stretch and compress and can shear off the concrete (appears yours did this) and if too stable it won't compress at all and may continue "pushing" the slab outward as the temp cycles. It would the ultimate if you could find a contractor that can do "rubberized hot pour" These guys know how to properly prep and place and are in and out. Gheck with the local road maintenance dept, or airport, When they have a contract going the workers love pick up work for beer money. You may want to look up cold pour also.
If it were my driveway I'd first remove the de-laminating sealant and re seal with a quality product made for the purpose (of expansion and contraction). I'd then create a few new transverse expansion joints where you are seeing the most movement. Say, 3 cuts spaced 25' apart. A wheeled concrete "wet saw" can be used to cut across the slab and put e.j. sealant in there. depending on the width of cut (I'd shoot for 1/2") it will take a couple accurately cut passes, or, have a contractor rout any single blade width cuts. Contractor rental yards will have these various tools ar pretty reasonable rate, and contractors should have them already. The transverse cleats could be placed where the e'j locations are if it's found they are required later.
Keeping the cracks sealed as you are is a great plan. Water intrusion will wash out the fines of the base sand/gravel and freeze thaw will accelerate concrete deterioation. Keep water draining away from the sides of the drive by grading to slope away.
Don't fret about all this info as I really don't think the solution is going to be big budget.