Doesn't a battery provide a current source since $V=IR$ and since $V$ is constant for a battery and $R$ is constant for a given circuit?

No- the meaning of ``ideal current source'' is that the device provides a constant current no matter what resistance you attach across it.

Similarly, an ideal voltage source provides a constant voltage no matter what you attach to it. If you change $R$, the battery (which provides constant $V$ no matter what $R$ is) will give you a different current. A battery is a voltage source, not a current source.

Some terminology: what you attach across a source is often referred to as a ``load''. Ideal sources are independent of load.