The leaky area in that second photo does *not* appear to be at the solder joint either.

Frozen pipes. Little doubt of that.

Our house here, for example, has hard water and 47 year old original copper. That stuff is still like new. Thick, solid, and not even close to being leaky (we know, because we've had to cut into it to add new lines during recent renovations).

On the other hand, it is somewhat grotty on the inside, so we're happy to have replaced sections of it with new, clean pipe.

In your case, I imagine the interior of the pipe is probably equally grotty, so just save yourself some stress, and consider this an upgrade to your drinking water quality.

Go with PEX -- less wallboard damage, considerably less labour, and the estimate from the plumber should be less -- if not, get another estimate or three.

The way things are going, by the time you sell again somebody will have determined that copper is as harmful to humans as it is to fish, and the PEX will be a prime selling point!

Cheers