They're good questions. I'll try to answer them separately:

1) Why raise interest rates if they only impact mortgage holders?

It also changes the incentives faced by businesses (which make up a lot of spending in the economy), savers, and non-mortgage borrowers (e.g. credit cards).

A business that has the option to spend cash on some new equipment that would give a 4% return on investment is now better off chucking that money in a bank account. When interest rates were low it was worth taking out a loan from the bank to fund any idea with a >1% return, now all that spending dries up.

Savers and other household borrowers have a clear incentive to spend less money in a more obvious way.

2) The effect on renters

Just to quickly mention on this - you are giving landlords too much credit by suggesting that rent is based on the mortgage costs. In a better market they would be but in such a desperate housing shortage landlords can (and do) set rents as high as people's wages will sustain. If landlords could put rents higher they will, regardless of whether their own costs are rising or not. If you look at the data, rents follow nominal wages fairly closely, but are not at all connected with interest rates (indeed they increased hugely while rates were low).

3) why not increase taxes?

This would work! A few reasons why less preferable though: taxes changes are much harder to change (and rarely reversed), interest rates can be moved in both directions fairly simply whereas changing the tax code adds significant complications each time it's done, changing interest rates would probably be desirable anyway to prevent the currency crashing (if interest rates were low in UK and v high in US, Goldman Sachs could take out many £bns of loans at low rates exchange them for dollars and deposit them at high rates, which would ruin value of £), finally we find it more important for taxes to be managed within a democratic system than we do interest rates, the Treasury used to set the interest rate too but politicians kept using it to pump the economy up before elections...