looking for feedback about a repair shop's claim

So my partner and I brought our vehicle into a dealership to get an oil
change and the spark plugs replaced. As soon as my partner picked up the
car after the work was completed, the manager claimed everything was
"fine" with the car, but when she walked up to the car she caught a strong
smell of burnt oil. As soon as she got in the car and turned it on, the car said
there was an issue with the clutch that required pulling over and stopping the
car. She did so, turned it off and back on again, and the warning didn't
reappear. In both instances, the burning oil smell and the clutch warning, she
gave the benefit of the doubt, and assumed that the burning oil smell was
due to some minor accidental spillage, and that the clutch warning was due
to the computer acting up in the wake of a diagnostic check.  She drove
home without incident and then back out to school the next day. A few hours
after she left I noticed that the smell was still lingering in the garage, I didn't
see any evidence of leakage on the garage floor where the car had been
parked, but was surprised that the smell was still lingering. My partner
arrived home that evening, and about 2 hours later we took the car out for
our grocery run. When we came back out from our last stop, the smell was
still so strong that we popped the hood to try to see if there was an evidence
of a spill that we could visually confirm. After not seeing anything obviously,
we closed the hood and started the car. Immediately the computer told us
that the battery wasn't charging and to turn off the vehicle. We took a chance
and drove home since we were only a quarter mile away, on the way the
engine indicated it was overheating. We made it home safely and the car
stayed in the garage, we called the dealership the next day and explained
the situation, had it towed to the dealership and it was left there over the
weekend. We fully expected that the issue was on their end, as the car had
been running fine up until we had first dropped it off for the initial oil change
and spark plug replacement. Now they are telling us that the smell is
because the clutch is going bad, and that the engine overheating is because
we need to replace our harmonic balancer and the drive belt. Is it possible
that if there was oil spilled during the oil change that the spill could have
damaged these components?  The car only has 45,000 miles on it, and the
clutch was fine before we took it in. When we first got it back, I did recognize
the clutch felt noticeably softer.  I'm concerned that they somehow damaged
something during the oil change and now are trying to charge us for damage
they did instead of admitting their mistake. Everything about the situation
feels off to me, but I don't have enough knowledge to really argue much
besides pointing out that the mechanic who checked out the car after the oil
change said everything was "fine" when it clearly wasn't, which tells me they
were dishonest from the start. Seeking any feedback and advice, thanks.