Is this a good method of limiting the current
I would say No, it is not a good idea to rely on the maximum rated current of the power supply.
Usually the maximum rated current of the power supply is the maximum current it is designed to handle, this does not say anything about the actual current which will flow.
I can design a 12V, 10A power supply and sell it as "rated for 1 A" (because I'm an idiot or just because I feel like it). Then you connect your, say 10V, 1 A LED and you assume that my supply will limit the current to 1A. But that will not happen. My supply supplies 12 V and can deliver 10A if it has to (even though I rated it for only 1A). So your LED will get 12 V and perhaps draw 5 A, your LED will then be damaged.
In your case using a 1 A supply with a 2 A load is also a bad idea. It is very well possible that the LED will ask 2 A and the supply will deliver what it can deliver, which might be more than 1 A.
Problem is, the supply is rated for 1 A and when you draw more than 1 A, anything can happen. It could safely supply the higher current. More likely the supply will be overloaded and therefore overheat and in the end it will suffer damage.
You cannot and should not rely on a 1 A supply actually monitoring the current and regulating such that only 1 A will flow. Only Lab supplies with current regulation can be expected to do this.
The general rule in the case of constant voltage is that the current rating of the supply should be the same or higher that the current required by the load (your LED).
If you had a 12 V 1 A constant current LED driver then indeed you can use that with a 12 V 2 A LED. The LED would simply get 1 A from the driver and not burn at full brightness. But you mentioned that the supply is constant voltage so this does not apply in your case.