The Tea Party is much more powerful in Europe than in the US. Read Ambrose Evans-Pritchard to see an example. It's just they don't call it the "Tea Party" in Europe. It doesn't seem to have a name over there, but that's what it is.
The basic philosophy underlying the Tea Party is, at its crudest: "We're not paying for them". Who's "we" and who's "them" differs across two centuries and across the Atlantic Ocean, but it's still recognisably the same philosophy. The Tea Party makes life difficult for the Lender of Last Resort, because the Tea Party wants some sort of guarantee it will get its money back. And that guarantee can never be made cast-iron. If it could, you probably wouldn't need a Lender of Last Resort in the first place.
We like to make a distinction between an entity that is illiquid and an entity that is insolvent. An illiquid entity can't pay what it owes right now. An insolvent entity can't ever pay back what it owes, even if you lend it some more to tide it over. Ideally, a Lender of Last Resort lends freely to entities that are illiquid but solvent. It lends enough to tide the entity over until it can pay what it owes.
Solvency refers to the future, and you can never know the future, so you can never know for sure that an entity is solvent.
Solvency also depends on the interest rate that the Lender of Last Resort will charge. At a high enough interest rate, the entity won't be able to pay what it owes, including what it owes to the Lender of Last Resort. At a low enough interest rate, the entity may be able to pay back what it owes. But that interest rate must also be high enough to compensate the Lender of Last Resort for the risk it undertakes. This feedback may be especially strong if the Lender of Last Resort is a central bank, so its monetary policy has macroeconomic consequences. So, even if we agree on the equations about the future, those equations may give more than one right answer.
And the Lender of Last Resort must act quickly, and be expected to act quickly and resolutely, even if it cannot quickly determine whether the entity is solvent, or what would be needed to make the risks worth the interest rate on the bridging loan.
This is not what the Tea Party wants to hear.
Despite my British North American bias, I have some sympathy for the Tea Party. The Lender of Last Resort function is perhaps antithetical to deliberative rule-bound democracy. It's a judgement call, and can't be anything else. If "we" are all in this together, so we sink or swim together, we can perhaps set aside these problems. If worst comes to worst, and it's a bail-out not a loan, we are only bailing out ourselves. But as soon as you start asking "Who, whom?", you never stop. Are Finnish taxpayers bailing out the Greek government, German banks that own greek bonds, French pensioners with funds in German banks, or Finns selling cellphones to those French pensioners?