If you have no check engine light, the PCM may not capture any "Rich" condition as it only occurs cold while in open loop. A lean condition has to bad enough and exist for a given number of seconds under certain conditions, spelled out on the diagnostic flow chart for a code as;
Enable Criteria. (The conditions it takes to run a code check, no other certain codes, or the test is aborted and actions taken by the PCM is the condition is detected plus the type of code. A or B.
The dealer will type in the VIN number and check. Some have 5/100k power train but emission is a different coverage.
NOTE: if you are close or "Yes" on emissions for a leaking exhaust manifold and the service advisor says, no...then ask for "Service Manger". The case can be requested to be reviewed by the GM regional service manager who visits each dealer once a month if the service manager agrees. They will submit a request and face to face meeting, set a date, but be prepared to bring in all service records from outside the dealer.
Now, a leaking exhaust manifold makes noise, but if air is allowed into the up-stream exhaust before "Bank 1 / O2 sensor #1...this can cause a rich condition. This will show up on a scan tool in block-learn for that side. Have it checked out for this info to "See" if this can be used as "Evidence". If true dual exhaust, the effected side will be a little blackened. Injectors on that side, will show longer open time to fix a lean condition. For example, good side shows the injectors open at 0.8ms warm at idle, but bad side the injectors are at 1.2ms. The O2 sensor will show higher voltage. No existing O2 sensor codes, either current or history. (TECH2 might be the only scan tool that will show up to 5 history data captures W /all engine PIDs. Do not clear codes as this erases everything that would help your case.
If you do your own service, go to rock auto, eBay or where ever you bought actual replacements parts, get printout or screen capture for show and tell. Now, even though this does not have anything to do with your broken bolt / leaking exhaust or cranked manifold, it still helps to have documents. You don't want to show up at the dealer with an external fuel filter covered in rust with OE markings. Even top tier fuel receipts, clean vehicle, good tire with even wear, oil condition, anything that shows neglect. This shows the regional service manager that you "Do" try to take care of the vehicle. They will ask if your fuel economy has dropped since the exhaust leak started and request documents like fuel receipts or your mileage note book. Yes, some people do that. I had a customer who flew a private jet for a corporation and took notes, photos and wrote down answer I gave him for questions he asked. Glad he moved!
Search internet for same year, model, engine and problem. Take a screen shop or use a camera that shows 20 or 30 of this problem. Even on forums like this one.
Any TSB that indicates broken / leaking exhaust manifolds for "X" vehicle thru "X" year range.
(NOTES)
TSB's are for techs to use to resolve a customers complaint, by using a new repair procedure, parts, bolts, manifold or even elongating holes. A "Re-Call" is a monitory fix and must be checked by VIN with the service advisor checking while typing repair order just for an oil change. If not done, the dealer will not get paid for any other warranty item on the ticket. A RECALL is either open or closed. It will show up at the dealer by VIN, regardless if your 2,000 miles from home.
If a RECALL is not fixed, while other warranty is done, the dealer has to present a signed document by the customer, showing that you where in a hurry, had to be a the doctors office soon, or in a parade in 12 minutes...etc.
Now, this is important, if the meeting is not in your favor, ask for split. Many times, the regional service manager will cover parts, but you pay labor...even thought its not 50/50. But ask for equal split, it doesn't hurt.
Many people quit and do not go that far. But, be prepared for thorough inspection of the actual problem, notes about aftermarket parts known to cause problems will be taken and be on the Techs' ticket. If you have a "Cold Air Intake", make sure the cotton air filter is not real oily as this effects the Mass Airflow Sensor.
(Last Step)
I was never a FORD dealer tech, but worked on enough to know I did not like them, but I was asked and become a stand-in FORD panel judge in cases where the owners felt the next step was a law suit and this was mediation by an impartial panel of Master Techs, Dealership Techs and others with motor vehicle industry ties. I don't know if they still do it, but it is an additional step in a process to keep a customer happy but keep out bad press.
A Local TV station may take it as a human interest story including their own go-to-car-guy. You know the one they go to when winter starts, they put him on a short TV segment to discuss low tire air pressure and wiper blades. A local dealer does not like bad press on the evening news.
(Just make sure the exhaust manifold is within the manufactures coverage before jumping)
Basically you have a number of common sense choices to follow. But remember, even a small claims court would require you to have a current ASE Master Tech stand next to you and speak, with all of the knowledge after he or she has put eyeballs on the problem completely. Photos, receipts mean a lot, but not parts. Do not attempt to bring used auto parts into court.
Lets say that over a decade ago, I have been the voice of official experienced car guy, a number of times, but only once in small claims court. But, W/O an ASE Master in small claims court to give positive testimony for you, you will loose every time.
Hope it works out for you.