This seems very unlikely compared to the obvious answer that they added a crappy antenna to keep their BoM and assembly costs low.

The radios on these are weak and adding an Omni antenna is exceptionally unlikely to violate EIRP.

Could this design accidentally radiate something it's picking up from the board outside of 2.4ghz? Sure.

But practically it's quite unlikely to in this context.

2.4GHz is already incredibly crowded and prone to interference (in urban environments). I remember once picking up emissions on a Bluetooth headset from a leaky microwave oven.

That isn't how this works. There is no change in emission power, the antenna doesn't add power.

Well mostly true.

You can replace an inefficient (turns power into heat) antenna with a more efficient antenna which will then radiate more.

And some frequencies are regulated by effective radiated power, where they basically care about the strength of the signal in the strongest location, so a more directional antenna could push you over the limit.