Any title insurance company will only process a claim if there has been a loss or an imminent danger of one such as an attack on the title, and only then if it is not of the insured’s doing or could have been prevented through action by the insured.
Since foreclosure is ostensibly always the insured’s fault (except of course in the rare case of forgery and intervening liens), you would be hard pressed to find any title insurance company that would see it as an insurable loss or attack on title.
The insured has an obligation to perform under the note they signed which usually includes a “successors and/or assigns” clause, and failing to follow through on that creates an uninsurable loss – securitization or no.
This issue would be after the foreclosing lender sells the property, not while the lender is in title. The obligation of the title insurer would be to defend an attack on the title for that insured. But the foreclosing lender would typically step in to defend and correct the foreclosure so no end loss to the title insurer.