Sure; but I'm unfamiliar with how a bailout of a specific business or industry would ever fall under that power. Do you have any examples?

If the business or industry is financial, a failure of that business or industry will have an impact on dollar-denominated assets which in turn can have an impact on the value of the dollar which the government can regulated. An instrumentality created by Congress that can do this is the Fed.

Example (going from memory here): after the 9-11 attacks, the Fed went into lender-of-last-resort mode in case there was a meltdown of the financial system.

Yes but this hazard was created by a previous violation of the constitution by de-linking the dollar from gold/silver.

If the dollar was backed by gold/silver this type of crisis would not be possible.

Article 1 Section 10:
"No State shall ... make any Thing but gold and silver Coin a Tender in Payment of Debts ..."

I know some have interpreted this to mean only "states" are limited by this and not the federal government; but it is a hard case to make due to the history of colonists with the Continental Dollar.

In 1798, Vice President Thomas Jefferson wrote that the federal government has no power "of making paper money or anything else a legal tender," and he advocated a constitutional amendment to enforce this principle by denying the federal government the power to borrow.

Article I Section 10 apparently allows the States to mint coins. Perhaps Texas can mint a silver peso and make it legal tender.

Jefferson was opposed to a government bank but lost on that issue.

The government has borrowed money and paid it back while we were on a gold standard. If Jefferson was opposed to paper money, why would he have not advocated an amendment banning paper money, rather than banning government debt?