A big question hasn't been answered yet: why was it leaking. Was it a defective line, defective inlet connector on the fridge, or sloppy installation work? It will make a difference in who is responsible for the damages. It goes without saying that anytime a water line is connected or reconnected to pretty much anything, it should be monitored for a while to check for leaks. A cursory check when it is first hooked up isn't enough. Water pressure can vary through the day due to demand or pressure tank cycling. You have to make sure the connection has been monitored for leaks long enough to catch the peak pressure period, especially if the house isn't fitted with a pressure reducing valve (PRV).

In addition to plumbing I have also been doing commercial refrigeration & HVAC for over 35 years so I like to believe my soldering & brazing skills are at least slightly better than the average homeowner/DIY'er and based upon the problems involved I would not attempt to put a tee on that bipass line. First off, it would be nearly impossible to get all the water out of those lines, yet if even a few drops of water are present you will not be able to heat the pipe enough to break one of those solder joints. Second, both the hardwater line & the conditioned line are very rigid so even if you could melt the solder you could not get enough movement in the pipe to separate the joint, but you could introduce some very interesting leaks that would be a nightmare to fix. Fortunately there is a much simpler solution. The bottom line in the first picture is the hardwater line. If you were to install a tee about 1/2 way between where that paper tag is hanging and the cabinet wall it would be fairly easy. (see attached illustration). Begin by turning the main water shutoff in the cabinet off and turning both of those valves to the off position then mark the fitting allowance on the pipe so you know how much pipe to cut out while making sure there is sufficient pipe to fit in the Tee. There is enough space between those two lines that you can get a Mini- pipe cutter in there. Next, using a strip of emery cloth, clean the ends of the pipe. You could solder a tee in there but given that you are not skilled at soldering, I would strongly suggest you use a sharkbite 3/4x3/4x1/2" reducing Tee. Note, in the illustration I showed the tee pointing down but if you need to go up you can also do that from this point. To install the tee the hardwater riser in the cabinet can be sprung back about an inch to give you clearance to get the tee in. (You may need a helper to reach the pipe in the cabinet). I would then solder a short piece of 1/2" pipe into a valve and once its cool insert that pipe into the side opening of the tee. At this point you could turn this valve off and open the mainwater shutoff valve and the water conditioner feed valve on and you will have to water restored to the house, so you could finish running the line to the hose bibb now, or do that at another time.

I think in your case, supply line hot would be the only option. For us, the old water heater was in the kitchen so we just transfered the wait time from the kitchen to the bathroom. We're having a tough time changing our water use habits. Prior to the new on-demand, turning on the shower and waiting seemed part of the routine. For handwashing after a potty stop, you knew to turn on the water when you entered! Now, in the kitchen the wait seems so long. A quick rince of dishes or a handwash doesn't seem worth the wait so everything but washing dishes is in cold water. I'm trying to figure out if the few times we need unlimited hot water (hot supply line) is worth paying for heating a small amount of hot water to sit in the pipes each time you use less than 3 gallons in the kitchen. I'm really leaning towards the cold supply line and seeing if 6 gallons will meet our needs. A dishwasher, which we do NOT currently own, uses 5-6. I've heard that hand washing wastes more water but I need to actually measure. Then the consideration is the cost. We thought we were going to heat water with gas and now are considering electric for the kitchen. I'll post what we finally decide to do.

Nothing wrong with flex pipe on a water heater. 99% of the problems come from people not knowing how to install it properly. If you look at a new water heaters supply and outlet nipples. You will notice a red and blue plastic insert. Those inserts are mainly for sealing the connections. Now when you use a nipple to connect two lines you do not have the plastic inserts to protect the rubber gasket in the hose connecter flange. Thus making them very easily to damage and causing leaks. Once the rubber gasket is damaged you can never get it to stop leaking no matter how tight you tighten it. Been using them for years and have had very little problems.