If the first law of thermodynamics ensures conservation of energy, why does it allow systems to lose energy?

“Conserved” doesn’t mean “never changes”. It means “this stuff is real, and the only way you have less or more is if some is taken away or added”. You can then follow that additions or subtractions.

Since your cold bottle has less energy, the conservation law says that energy has not disappeared, it’s just gone somewhere. You can find it. You can figure out how it got there.

I'm not sure why people are saying this only applies to closed systems. This law actually applies to all systems. The first law is conservation of energy. It says the change in energy is equal to the energy that enters/leaves it in the form of work or heat. i.e. $$\Delta U=W+Q$$ where $U$ is the internal energy, $W$ is the work done on the system, and $Q$ is the heat that enters the system. This equation essentially just says we can track where the energy of our system is coming from/going to. It isn't suddenly appearing from or disappearing to some "unknown nowhere". It's energy conservation.

In your system, heat left the system, and the system did some work on the environment I suppose (though this might be negligible). In any case, $Q<0$ and $W<0$, so it should be no surprise that $\Delta U<0$. Energy has left your system (and has gone somewhere else), so the internal energy has decreased. Energy conservation is true, and it's present in the first law here.

Conserved here doesn't mean that it is conserved only for your system (Not unless it's an isolated system). It is conserved for the whole universe. The total energy is constant. In performing any work or task , all you are doing is taking some energy from the surrounding and giving it to the system or vice versa. These 2 effects balance out or cancel each other if you consider the whole universe making thermodynamics "Perfectly balanced, as all things should be"