I would not panic. This is one of those situations were it would be prudent to consider replacing the line within a short time, but the question then becomes, what is a short time when discussing a water line?

I would consider replacing it within the next 3 to 5 years, but you need to get the project going now.

First of all, this line is serving as the municipal main for 6 homes, so the cost of replacement should be equally shared by all 6 homeowners. (From my experience you will always have at least one homeowner who balks at the idea, but not to worry, there are laws compelling them to pay their fair share, it just causes delays and increases the cost when you have to get lawyers & courts involved).

Obviously copper has proven to be a good choice, but with the price of copper today it would be nearly cost prohibitive. A much better choice would be HDPE pipe (high density Polyethylene pipe), which is what is being used by most municipal water suppliers today.

The next question is, how will it be installed? The obvious solution would be to get a backhoe and start at one end of the street and dig a trench to the other end, but that introduces a bunch of problems. What about all the sewer & gas lines that are in the easement? And all the house sewer and house gas lines that are tapped into the mains? In a municipal area you may also be confronted with underground telephone, cable, commercial communications lines, data network lines , storm sewers, and who knows what else is buried in that easement. And that is just assuming that all the houses are on the same side of the street. What if three of the houses are on one side and three on the other. How do you replace the line under the street and still maintain a traffic flow so the ppl can get to their house during the construction?

There is a way. Instead of digging an open trench they can use a directional horizontal drilling machine.
A horizontal drilling machine is set up at one end of the run and a drill pipe is sent down to the required depth then they turn it horizontal and drill under all the existing lines without disturbing them. The tip of the horizontal drill bit has a tiny radioactive chip that can be followed by a hand held receiving unit while standing on the ground so the ground hand knows the exact location and depth of the drill. As the line passes each house the ground hand can mark the exact location where the underground pipe is passing so a small access hole can be dug to make final connection from the main line to the house line, and if you have to run lines to the other side of the street they can drill under the street without disturbing the street or traffic flow. When the steel drill pipe reaches the end of the run they turn it up so it will come out above grade. The plastic HDPE pipe is then attached to the steel drill pipe and as the drill pipe is pulled back it will pull the HDPE pipe into place.
The HDPE pipe is available in rolls up to 1,000' long so you will have no underground joints except for the individual taps where each house ties in.

Now in regards to joints. HDPE can either be joined by mechanical connectors or they can thermally weld it.

You also need to consider the size of that line. Back in 1929 when that 2" copper was installed all houses were connected with a 3/4" line, which was fine for the needs in those days, but consider this. In 1929 each house had one bathroom with the simple layout, WC, tub & lav, a kitchen sink and a laundry tub, where they operated a wringer washer about once a week. And when they did laundry the washer was filled with about 10gallon of water once, and they ran 4 or 5 loads through the same water and another 8 or 10 gallon in the rinse tub. Today our washing machines consume 5 to 8 gal, two or three times per load.

Today everybody showers daily, but back in 1929 they showered faithfully on Saturday night and sometimes once or twice during the week. Then again, how many of those houses have had additions with additional bathrooms and dishwashers in the kitchen? And that doesn't even consider that ppl are much more inclined to water their lawns or wash the cars a couple times a week today. Typically we use about 3 times as much water as they did in 1929. Now you may argue that you are doing all those things now and the 2" line is doing fine, and you would be absolutely correct. The volume of flow is not restricted by the line size, but as the demand increases the velocity of flow through the line increases. The problem is that as velocity increases proportionally, the pressure "friction head loss" increases exponentially.

Hopefully, when you replace the line you will not design it just for the immediate needs today, but rather allow some expansion for what may come about in the next 83 years.

When comparing the capacity of lines the formula is "Large diameter squared divided by the small diameter squared" will yeald how many small lines have the same cross sectional area as the large line.

Let us consider your existing 2" line with 3/4" lines running to each house.
(2x2)/ (.75x.75)= 4/0.5625= 7.1.
That means the 2" line has the same cross sectional area as 7.1 lines that are 3/4", which was fine for 6 houses with minimal fixtures.

Now there is a rule of thumb in the plumbing trade that if we increase the size of a line by one nominal trade size it approximately doubles the volume of flow.

Let us see what happens if you increase the size of the pipe one nominal trade size from 2" to 2.5".
(2.5"x2.5")/(.75"x.75")= 6.25 / .5625 = 11.1
As you can see, simply by increasing the size of the line from 2" to 2.5" you nearly double the cross sectional area, which means you can have nearly double the volume without causing a substantial increase in velocity.

Simply increasing to 2.5" should net you a substantial pressure increase immediately and your system will have provision for additional loads as lifestyles change in future

Often more effective in concept than actuality.

Horizontal boring has resulted in numerous cases of damage to existing utilities of all types as use has exploded to try and limit open trenching.

You need to start by doing research about the level of documentation available for the existing utility lines.

It could be anything from adequate to almost non-existent.
Especially on a very old privet road section.

Good luck.