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The item with the disconnected vacuum hose is an EGR valve.
If working and connected correctly, it will come on at speeds of 30 to 40 miles an hour or higher on a warmed up engine. It enrichens the air/fuel mixture during those events.
There should be zero EGR operation at idle.
With it disconnected the engine may ping or spark knock during those events. If the engine does not ping or spark knock the air/fuel mixture has been adjusted too rich.

It does sound like the carb needs to be rebuilt or replaced. Going to an aftermarket carb may mean connecting certain pieces in a makeshift manner.
Accelerator
Choke
Transmission down shift

The Amazon carb looks interesting. Because the price is so low I might be inclined to buy it while I am getting the original carb rebuilt.
I would agree with you that buying a rebuilt carb is a hit or miss type of thing. Depends on the cleaning the carb gets and the skill level of the person reassembling.

I personally do not have issues with an electric choke if the electric connection can be made easily. It is a more accurate choke than non-electric.

I suspect your vehicle has other issues besides the carburetor. You have had it a long time but do not drive it much. The EGR should be addressed.
I suspect the intake system is gummed up and there may be amounts of varnish like coating on the inside of the intake and intake valves.
You may have leaking vacuum lines. (Old rubber cracking)
A diagnostic step would be to get vacuum gauge readings on a warmed up engine.

When was the fuel filter replaced last?

If it were me ....
I would make sure all of the maintenance is current. (Filters, ignition system, adjustments)
Run engine cleaner through the fuel system.
Get the EGR system functional. Remove the spacer plate and valve. Clean plate and test valve.
Get a vacuum reading and repair any vacuum leaking issues
Rebuilt the existing carb - cleaning is critical. The internal passages are most important.
Buy the Amazon carb and run it to see if that is a good solution while the existing carb is being rebuilt

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I assume the numbers you posted for intake vacuum were in inches of mercury.
If correct, a 21 to 22 inches of mercury reading and a steady needle is a very healthy engine.
And the vacuum readings based on throttle operation are spot on.

For the vacuum lines, is there a paper sticker on the front of one of the valve covers?
If yes post a picture.
I may have an engine vacuum diagram I can post for you.

The function of an EGR valve is to reduce some of the NOx. It takes some of the exhaust gas and re-routes it back into the intake.
This is because not all of the fuel is burned. (Versus modern electronic engine controls)
However the engine can not handle recycled exhaust gas all of the time.
So restrictions on EGR operation are only on a warmed up engine and only at speed. Like the vehicle is going 30 to 40 MPH or faster.
EGR will not reduce fuel economy. If anything, it improves fuel economy as the mixture can be slightly leaner because the fuel in the exhaust is being burned.
It is possible performance is less. Only because the unburned fuel in the exhaust is not that good of quality.
However that pales compared to the lowered cylinder compression and the reduced performance of the cam and timing chain.

The combo electric choke, yeah Ford used a lot of variations. The goal is to open the choke as fast as possible and keep the choke fully open after that point. UNless the temps are below freezing, choke operation is only needed for 15 to 20 seconds on a cold start.

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IMO the Marvel Mystery oil is not needed. It won't hurt. But may not do much if anything either.

That sticker the numbers and lettering is coming off is the one I posted about.
However I think the same data is in the lower right of the pic you took.
Should start with a 7. The rest is not legible.

This is the Calibration Code. Or at least this is what I am looking for.
With that code I can look up in an emissions book for engine vacuum routing.
Typically there are about 30 to 40 calibrations for any given year back in that era.
I have a number of books and pretty sure I have model year 1977
Hopefully I have your exact calibration.

See if you can carefully clean the sticker you posted as it is more durable than the one on the front of the valve cover.

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You might need to carefully wipe the paper tag. It is the far right side of the tag that is important.

I have attached a pdf of an example of the calibration sheet. Originally these are 11 X 17 in size and on colored paper. The one I attached is orange and I have had to copy it and reduce the size so it would be legible and fit into my scanner.

The calibration code is on the bottom center and this example it is 7-76J R11
How that breaks down is -
First digit 7 means this is when the calibration was released. 1977
The second grouping 76J is for the specific engine group and application. 76 is for 351W engines
The last grouping is revisions. Starts with R00 which is blank and R10, R11, R12
Of all of the groupings the revision code is least important.

When I first saw the 7 (not the other numbers) on the valve cover sticker you posted I was thinking that was it! It is not. And not sure what 7-304 means.

BTW 7-76J R11 (which I have attached) is for a 1977 351W engine in a Econoline with under 6000 pounds GVWR for California only. That is how specific these can get.

I do not have all the sheets for every year. I do have a lot of them. The book for just 1977 & 1978 is about two or three inches thick.

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The punch out section of the paper tag is when the engine was completed at the engine assembly plant.
E10 would have been May 10th of that year,

It is the far right that starts with "C" at the top
What follows is a number that starts with a 3. That may be enough.
Next box down would be R and ???
Last box would start with an S and is a service level designation used in the assembly plant

Charcoal canister -
The blue cap is a vent.
The black cardboard corrugated hose goes to the air cleaner.
The vacuum line goes to the carb, I think. It may have an inline valve.

The metal can in the drawing I posted is an EGR vacuum reservoir. It ties into an EGR amplifier on the top right rear of the engine.

I will pour through the book.

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Full sized LTD correct?

And which transmission?
C4 or C6

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OK pretty sure I know your calibration code assuming you have full sized 1977 LTD and not LTD II
It is 7-32B R10
This is the only calibration for 1977 full sized Ford with 351W 2V and automatic transmission.
That code looks like it fits for your sticker.

So much for the good news. The not so good news is, I do not have that calibration sheet.
The sheets were sent in batches to the dealership during the model year. For model year 1977, I only have the 5th and 6th printings.

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I am going to do some more digging to see if I have more. I have a shed full of publications.

The cardboard hose was never a tight connection

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Just letting you know that I have not forgotten you.
My "library" is a termite infested wooden shed that leaks. All of the publications I have are in boxes. I have at least a couple of dozen boxes. Over the years the boxes have gotten moved to get away from roof leaks. As such the boxes. have lost any logical order long ago.

Still digging

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