Treasury Secretary Yellen said this would be a "constitutional crisis". What would that entail? Would it be up to SCOTUS to tell the executive it can't do that? And then SCOTUS would what... tell the executive what the priorities must be for paying its expenses? Or just leave that unspecified? I imagine the executive could continue to defy SCOTUS by simply arguing that they must do so to ensure that the Treasury pays for the spending that Congress has already authorized.
If they follow the 2011 plan, it would essentially mean that the government will start jerking like a car with a faulty transmission, because [under that plan] priority will be given to paying debts rather than agencies, on a daily basis.
One cannot predict how Treasury will operate when the debt limit binds, given that this would be unprecedented. Treasury did have a contingency plan in place in 2011 when the country faced a similar situation, and it seems likely that Treasury would follow the contours of that plan if the debt limit were to bind this year. Under the 2011 plan, there would be no default on Treasury securities. Treasury would continue to pay interest on those Treasury securities as it comes due. And, as securities mature, Treasury would pay that principal by auctioning new securities for the same amount (and thus not increasing the overall stock of debt held by the public). Treasury would delay payments for all other obligations until it had at least enough cash to pay a full day’s obligations. In other words, it will delay payments to agencies, contractors, Social Security beneficiaries, and Medicare providers rather than attempting to pick and choose which payments to make that are due on a given day.
As further discussed in there, it's not quite like a government shutdown, because employees can't be furloughed, but they'd have to work for IOUs essentially, rather than getting paid on time. Stuff like this happens/happened in some countries, usually described in derogatory words in the US.
That will also likely result in numerous lawsuits, potentially with contradictory lower court decisions as to what the gov't should pay first, eventually having to be reconciled by the SCOTUS.
So, it's also possible that it won't go entirely according to plan, thus short-term technical defaults on Treasuries are also possible, as in fact did happen in 1979. How much the credit rating of the US would be affected depends on the magnitude of these events.
And if various officials start ignoring court orders, they can be jailed for contempt, although in some of those aforementioned countries, police also routinely claims it didn't receive any such orders, when the person to be jailed is some potentate.
FWTW, there's already a lawsuit (filed by a union of government employees) calling for Yellen to ignore the "unconstitutional" debt limit. As that piece discusses, the Treasury might actually welcome "losing" that suit. (Like it happened with one of the mifepristone lawsuits against the FDA, for that matter--the one which ordered them to keep providing the drug, albeit only as an injunction.) That basically would create legal cover for the Treasury, at least temporarily. (Whether the debt-limit unconstitutionality arguments have a lot basis, I won't try to discuss here, but a quick search does find some paper penned by some law professors advocating that. But it may be more difficult for Congresspersons to 'sue back', as courts have been somewhat more inclined to limit their standing. Still, it's not exactly impossible, if a Congressperson can convince a court their vote has been "nullified" for instance, which in this case doesn't seem so implausible, as many of the Congresspersons who voted for the last debt limit are surely still alive/around.) The scenario advanced in the Q, that Yellen would just ignore the limit (with no other intervening events) seems the least plausible to me, so take this answer mostly as a frame challenge.