We had something similar done recently. I had an idea very similar to what you're thinking -- to push PVC through and out to the main sewer hookup. A plumber told me that wouldn't be legal. That makes sense, since a smaller pipe pushed through would not yield a sealed drain connection. There would probably end up being sewage flowing outside from the end of the pipe.
What we have are 2 or 3 foot sections of clay pipe. They had shifted a bit and were getting clogged with roots periodically. In the process of getting it fixed I learned something interesting: The sewer cleaning people who were charging $300 to ream out the roots only use a small bit to make a hole. They don't actually clear all the roots from the inside of the pipe. As a result, that needs to be done every
6 months or so. For anyone doing that a better solution might be to get a commercial grade, 50' reamer with cutting bits and maintain the drain themselves. Such a tool is about $500, I think, so it would pay for itself on the second use, and the reaming would last much longer than one gets from the sewer cleaning people.
The final solution was overpriced. It was $4,600 for a crew of about 8 men, but most of those men were not necessary, and most of them weren't actually doing anything aside from handing a wrench to the boss. It took a few hours, then one man stayed for a few hours more. But the price is typical. It's based on a rate of $100 per foot. The solution is clever. They first run a sleeve into the old pipe, after having thoroughly reamed out all the roots with a number of different cutting bits. The sleeve then somehow gets pumped with a liquid. I don't exactly know the details. Then they inflate the inner sleeve to push the liquid against the pipe and they cure the liquid, with heat, I think. The result is a tough plastic pipe that looks to be maybe 3/16" thick, which is molded to the old clay drain pipe. No more roots because they no longer sense the water. As part of the deal we got a DVD movie showing the length of the finished drain pipe. :)
It seemed like a very good solution, though it's only been there a year so far. I think the companies are price gouging, but probably because they can get away with it. To open up the ground and run new pipe to the street costs in the 10s of thousands, so the liner is cheap by comparison. And of course these are plumbers getting plumber's rates.
One limitation, though: In our case the clay pipes had separated a bit and the sealant between them was gone, but the pipes were still OK. In some cases the old pipes will be collapsed, badly broken, or rotted away. In that case I'm not sure they can use the plastic liner. The whole thing would probably have to be ddug out and re-piped.
On the supply line, I don't know. If you're hooking up to municipal supply you're going to have to talk to them to see what they allow.