Is increasing housing scarcity more inflationary than increasing scarcity of other goods, due to debt?

My understanding is that in general, increasing scarcity of a good is inflationary, because it leads to price increases. For example, current situation with oil.

Similarly, if housing is increasingly scarce (e.g. due to population growth and supply constraints), then prices increase. In addition, the price increases are often paid by consumers with mortgage debt, which also creates money.

My question is: does the fact that price increases for housing are paid with debt add to inflation, and is there an economic theory which addresses this? And does a tendency towards large mortgages (relative to income) lead structurally to inflation?

(The specific context I am thinking of is Australia, which has had extreme housing price increases over the past few decades, and it is common to borrow large multiples of income for mortgages with small deposits; and also currently is having some problems with inflation).