That is indeed a gentlemanly and fair thing to do... however, certainly
in the UK, you are told that you must NEVER admit liability, simply
exchange the required details and let the ins cos sort it out.

This happened to a friend of mine except they hit the brake pedal
rather than the clutch (due to a un-diagnosed medical condition I might
add).  The police arrived as the accident was holding up traffic.  The
other driver (the one who rear-ended my friend) had said that "they
were expecting her to be out of the way by the time they made their
exit (from the roundabout) so they were actually accelerating at the
time"  but then went on to complain to the police that my friend should
be cautioned for bad driving etc etc.  The policeman quietly took her
(the other driver) aside and said "I think you should keep quiet
because you'll probably find that you'll be held responsible and we
might actually bring a care and attention case against you".

The insurance company wrote to my friend to say that they (her own
insurance) wanted to accept a 50-50 liability.  Despite having legal
cover etc they weren't going to pursue a full claim against the other
driver.  My friend and I then wrote back quoting references from the
highway code etc and in the end after about a year the insurance people
finally sorted it out.

This is where I add insurance companies to my "cowboy" list as they
were clearly more interested in simply closing the case rather than
fighting it.

His stalling is something you're supposed to anticipate, especially since in
theory, you aren't even moving at the time. I do not think yoiur insurance
carrier would have changed their position on paying the claim.

Same story, but this time the guy tried to blame the hittee. I don't
even think "the you stopped quickly argument" is an excuse especially
if you weren't moving at speed or nothing pulling out of a junction.
I recall during my driving tuition, my instructor explained that most
people at a junction/intersection only see the other car in front jerk
before taking their eyes of it to see if there is any on coming traffic
so they can join the traffic and that if they were looking at the guy
in front they wouldn't rearend them, which I'm almost sure is the case
for you.
A stalled car, a kid jumping in front of you, a fast oncoming vehicle
its all the same, thats the reason for the emergency stop manuever, and
if you hit that guy in front then you certainly were not paying
attention especially at that speed.
The hitter even tried to blame the hittee, claiming he stopped too
quickly, at which point he pointed out that he looked saw no oncoming
vehicle, started to move, looked again and saw an oncoming vehicle so
braked. Getting hit was actually driving without paying attention to
the traffic in front.

As for insurance details, get it even if its a scratch, the excitement
of an accident means you don't notice anything, in my friends case,
only upon opening the boot and looking inside did the damage become
apparent, at which point the boot wouldn't close.
If you feel the car was hit, then the car is alot worse than even you
think, get the insurance details.

The best example of this I read was a police report, where having pursued a
vehicle the driver decamped and absconded on foot.  Did the police car run
into the back of the stolen car when it stopped very suddenly - well no the
escaping driver didn't engage the handbrake so the car rolled back up hill
into the police car...

Other examples include the train driver dazzled by the sun in his eyes as he
was making an evening run into Paddington, etc. etc.