You don't fail an emissions test because there is too much carbon inside the exhaust system. There is too much carbon/sooting inside the exhaust system because the carb is running grossly rich. The gross rich carb will fail the emissions test - not the resulting carbon in the exhaust. You fix the problem by fixing the source of the carbon sooting: Fix the carb, and make it run right, and the correct-running carb will then cause the carbon sooting to burn out of the exhaust after you do a couple of wide-open-throttle runs to blow out the big chunks. Fix the source of the problem - not the results of the problem - and the problem will go away. Fix your carb and timing. You can clean out the inside of your exhaust system all you want, but the car will still fail emissions with a rich-running carb that caused the carbon in the first place.

 

The HC tailpipe test is likely the one you are referring to. That is the Hydro-Carbon emission test. Basically un-burnt fuel. The original requirements or upper limits for those years were a max of 200-400 HC. It is not the black particles coming out that gets tested, it is the amount of gas molecules in the air. There was no particle restrictions when these cars were new. But the air (gas) emissions regulations were brand new for 1970.

I squeeked my 70LT-1 thru NJs state run emission testing for about 10 years. I had to retard the timing to near zero, pull the vacuum advance, and lean out the carb so much it would barely run. We did this on an emissions meter at a buddy's shop.

Then I ran it thru the state test line and it would pass.

But it ran so badly that way, that I would pull over only 1 block away, in a shopping mall parking lot, and reset the tune-up, so I could drive it home, and it would run right, without stalling, bucking, etc.

The early 70s were a real learning curve for the car manufacturer's and emissions. Try this: Lean out the carb mixture screws until you get a 20 or even 50rpm drop at idle, and leave them there. Disconnect & plug the vacuum advance. Set the idle timing to no more than 4*. Set idle rpm to factory spec, 700 or maybe 800rpm. With all fresh spark plugs, wires, etc. That should do it.

Those emission controls did not really work that well, or run that well, and 2 years later in 1975 they had to resort to catalytic convertors.

Your 73 probably has an EGR valve. That was the first year for that. You need to make sure it is working, if they test for NOX. Push it open slightly with your finger at idle, to make sure it affects the idle, Then let go and idle should return to normal. It must be closed at idle. But if you rev the engine up, you should feel it open, with your finger. You may need to remove it, and unclog the passages in the intake, they clog with carbon. If it does not affect the idle, they are clogged. If they only do an idle test, standing still, it will not matter, they will not test for NOX. But if they do the 20mph rolling dynomometer "driving" test, then it has to work correctly. We had both versions of that over the years. The dyno test version was much harder to pass.

If your nose can smell gasoline in the car's exhaust (it stinks) then you are way over the limit for HC. Likely 1000 or 2000 ppm HC.

 

An excellent way to clean the carbon deposits out of the cylinders is to use water, or GM top engine cleaner.
The EGR valve engines get a lot of carbon deposits on the pistons, and they glow red, and effect the emissions, a little bit anyway.

Best video I could find:

We did this a LOT in the 80s on those old cars, it works really well.
But on a carb engine, the water just goes straight into the carb.
Rev it to 2500-3000 and add a steady small stream of water, like he did, maybe 1/8" (2-3mm) wide, and if the engine slows down, stop the flow, till it recovers.
Proceed for like a minute or so.
DO NOT STALL THE ENGINE!!!! Or you could hydrolock it and ruin it.
I do not like using the vacuum hose method because it is more difficult to control the flow.

When you are done, go out for a drive, and give it a few long full throttle blasts to clean the loose carbon out.
We called this the Italian tune-up.
It really works. Don't laugh.
We were so high tech back then!