inductance - How can the infinite reactance of either primary or secondary in an ideal transformer draw current?

Assumption 1: An "ideal" transformer is said to have very large primary, secondary, and mutual reactance.. (self-inductance/mutual-inductance tending toward infinity), has a unity coupling coefficient (zero leakage flux), High or infinite magnetic permeability, absorbs zero real power (is lossless, 100% efficient).

Assumption 2: From a pure circuit analytical and mathematical standpoint, and without the "real" model elements, the infinite primary and secondary inductances in the "ideal" transformer will draw current when secondary load is not open, and zero current when the secondary load is open or tends to infinity.

Problem 1: How can the infinite reactance of either primary or secondary draw the current in assumption 2?

Problem 2: The secondary load gets transformed and appears in parallel to the primary inductance, so if the primary reactance is virtually open, why even put it in the circuit? what good does this do?...there are an infinite amount of parallel opens in any given circuit.

Thanks in advance!