Dominate? I doubt it, although the chassis systems are making both great strides and headway in the world of shooting whether you decide on hunting or targets.
This is also one of those contentious topics from the standpoint that everyone here has very definitive ideas about what is right and what does not fit into their own little world of acceptable. It's no different than asking about the .30-06 Springfield and any other cartridge as to which is the best. Or .45 ACP vs. 9mm in pistols.
I started in custom, handmade walnut stocks, fine English, Circassian, French, Turkish or any other flavor which caught the eye of a client. There is simply no equal when it comes to the exceptional beauty of a truly gorgeous piece of walnut transformed into a perfectly executed and fitted stock. The drawbacks though are cost and time. None of us can afford to work for free and these stocks take lots of time and care to complete. There is no way around it.
Then along came the synthetic stocks, a God-send for extreme environments and stability. Great for ease and simplicity of fitting with very flexible constraints on actions and barrel combination. And best of all, they were lots less expensive than the handmade, wood alternatives. They required pillars and bedding, of course, then the application of any other accouterment. We even got what is referred to as a chassis installation in the form of a bedding block and DBM bottom metal/trigger guard, along with the ever-increasing price tag. Some of these stocks can get to $1,500 when you add in the folding stock option.
Then came the plywood generation. Yes, laminates, the polite reference to plywood, came onto the scene and were immediately labeled as being cheap and ugly. Cheap they were and ugly they were and still are but the general shooting public has acceded to the acceptance of this material in place of the rising costs of synthetics and walnut because they are cheap. Gunsmiths can inlet and bed them the same as the synthetics but the overhead is slightly less. The greatest downfall currently is the rapid escalation of the costs of producing the plywood target stocks for competition. These are nearing the point of insanity when the carvers are asking for and getting the high side of $1,500.00 for plywood. Silly in my opinion but they are getting it. Remember what I stated about opinions?
Now we're in the mechanized age of chassis stocks. Virtually everyone is familiar with the AR platforms and have become comfortable with their association with better firearms. CNC programming and the spread of CNC machining into even the smallest of shops has added a huge impetus to the marketing and sales of the chassis. Everyone with the money and skills necessary to operate a CNC mill has entered their personal expression of their concept of a rifle stock, executed in aluminum, into the market. We are blessed with a wide range of not only designs but levels of costs and accessories. It can get to be lots of fun to design your own idea of a stock just by the manipulation of some menus provided by the manufacturers. And voila! A few weeks later your chassis shows up ready for the drop in fitting of your barreled action. Some even state that bedding is not required under any circumstances.
If the chassis you desire is not in inventory, the delay is usually just a few weeks. Try that with our synthetic stock makers these days. The delay is measured in months, not weeks. To be fair though, our synthetic stock makers are making inroads to decreasing the wait time by several methods, so some wait times are coming down.
Is the chassis everyone's cup of tea? Not hardly. See the above about opinions. They will never have an English-style stock at the wrist. But you can add wood in the form of grips and side panels to offset the feel of cold aluminum. The chassis will remain the most adjustable of all stock options, easily managed by the owner with simple hand tools, eliminating the gunsmith for this task. The chassis itself practically removes the gunsmith from the equation of rifle building due to the simplicity of the designs. Add in the growing demand for the 'REMAGE' barreled actions and you can call yourself a gunsmith and skip all the schooling and practical training and wait times.
I have tested, manipulated and competed with almost every chassis available. Some can be adjusted to fit me easily and are very comfortable to shoot. Others fall short, not because they are inferior, but because my shape does not lend itself to the design of those chassis in particular. I have a couple which can be purchased for $400.00 while I know for a fact that there are designs out there running in the $1,500 - $2,000 range, given the selection of add-ons. There are some exceeding this top price also, much of which can attributed to being manufactured outside the U.S. or through the use of premium materials. The top dawgs in the field keep their prices high in order to claim that top spot due to the premium cost of their designs. I don't fault them as I own a couple but you will hear all the complaints and derision from many who simply don't comprehend why they are expensive.
This can be a much longer discussion, touching on more of the subtleties of design and manufacturing but that would take a long time. I like and use all of the various materials for stocks, yes, even some 'plywood'. My all-time favorite will always be fine walnut. The others though will be in the inventory and used much more frequently these days.