Answer: If you have to "feather" (lightly tap) the gas pedal to keep the engine from stalling, the most likely problem is the accelerator pump in the carburetor. It is common for these to fail and you can replace them without having to replace or rebuild the entire carburetor. The two leaks are easily remedied with new gaskets.
That being said, your mechanic is probably giving you good advice, although I would have the carburetor rebuilt rather than replaced. He has to remove it from the car to replace the base gasket, and the accelerator pump likely needs to be replaced, so rebuilding it doesn't cost much more.
It sounds like the places that you’ve found on the internet are going to restore your carburetor, rather than rebuild it. This is all good (and expensive) if you're planning on having the car judged at high levels. However, if you’re going to drive the car, and occasionally enter it in local shows, you simply need to have it rebuilt by a reputable shop. It will be spotlessly clean, it will come with a new base gasket, and it will only need minor adjustments.