Can water hammer be fixed by replacing a tap?
My answer is based on this:

I have a mixer tap in the kitchen that has been causing an INCREASING amount of water hammer

It sounds like the plumber properly diagnosed the issue and can see that the valve is worn out. The more you use it then the more worn out it gets thus explaining the increasing amount of water hammer.
If the issue can be solved without a water hammer arrestor then I would go for that solution. Consider the pros of not installing an arrestor:

One or more fewer points of failure

The tap is required, an arrestor is an accessory


Arrestors can fail for various reasons and the hammer will return

If you install an arrestor then you're only hiding the problem.
Yes, the arrestor will absorb the noise but all that water force is still being exerted in your pipes. If you're only taking care of the symptom then the water hammer can still cause joints to fail and then you'll have a much bigger problem on your hands.