With all due respect you are wrong - if the first mortgage forecloses junior lienholders are invalidated. I use the term junior becasue even though taxes come AFTER the foreclosing deed of trust they are NOT junior lien holders.
I speak from experience of buying foreclosures on the court house steps right here in Williamson County, TN.
Your answer is mroe reflective of what MIGHT happen in a short sale where a junior lienholder can refuse an amount they are offered "short".
When buying AT the foreclosure auction there is no financing, no title insurance, no warrantability, no inspections - zero - it is very much buyer beware and yes they want your cash straight away.