The short answer is that:
- Yes a well sorted road bike can increase your speed, if it is done right;
- That said, you can optimize a CX bike to be very similar in speed to a dedicated road bike; and
- You need to get a number of pieces right (regardless of the bike category) to make higher speeds easier.
While a human on a bicycle is an incredibly efficient transportation machine we don't have much power to work with, so speed gains must be made through a focus on efficiency. You can of course always become a stronger rider, but even the strongest riders are limited by a slow setup.
For context I commute 15 miles (24 km) each way to work (48 km total) with a weekly average speed of about 20 miles per hour (32 kph). This is over mostly flat terrain (only 110 m climbing). I usually try and maintain about 21-26 mph (35-42 kph) on the flats to hit that average seed. This is on a "performance endurance" road bike.
In previous years with a more standard commuter setup and I was on average about 4-10 kph slower depending on the setup (slowest was on wide commuter tires with fenders and a very upright position). This was for a similar level of fitness and power input (measured with a power meter).
That is a gain of 14-40% from paying attention to all the little details and I must say I am enjoying my commute much more!
In terms of average speeds, it really can be a death by a thousand cuts. For a commuting setup I have listed the pieces I believe gave the biggest speed gains. No single one will be overly noticeable on its own, but add together it can translate into some considerable gains.
Setup details I believe that had a large impact include:
- Panniers - like pulling sails through the wind. I now use a small roll bag mounted to the front bars (backpack would be faster, but I don't like carrying things on my back).
- Body position - Upright body postures may be more comfortable* but it increases your frontal area (i.e., how much wind you push). I have been fixing flexibility issues and have dropped my cockpit a couple inches down and extended it by an inch. This lets you present less of your chest to the wind (i.e., smaller frontal area). Width of your handle bar also impacts your frontal area. I keep the width the same as my shoulders which actually feels best on a low slung cockpit.
- Tires - Puncture proof tires roll slow. I am running performance tires set up tubeless so that I can still get some puncture resistance while going fast. This provided a really noticeable reduction in rolling resistance.
- Wheels - regular box section wheels add noticeable drag at higher speeds, I went to some moderately aero wheels this year (and matched the correct tire width to the rim for least aero drag).
- Clothing - anything even slightly baggy is like dragging along little speed sucking parachutes. I once again wear tighter road clothing as I used to many years ago.
* Note: I understand that lower body positions are usually perceived to be uncomfortable, but find lower body positions fine under higher efforts and at times find higher positions uncomfortable for for harder efforts as you keep bending your upper back to get out of the wind, rather than stretching it out on a lower and longer cockpit. Optimal position really depends on the type of riding you do.
If you notice all of the optimizations I mentioned may be equally applied to a road or CX bike. So I would start there rather than necessarily a new bike (n+1 rule be damned). What type of bike is only a detail in the quest for speed!
There are also other basics (such as saddle height) that I have assumed you have already figured out (i.e., see mattnz's answer).
Finally, while not directly attributed to the bike setup other factors that can affect the ability of your body to do work:
- Sleep, eating, physiology (e.g., hydration) and endocrinology (e.g., hormone imbalances).
You need to have all these in order as well to maintain high average speeds.