suggest a compromise
Perhaps use warm water for everything?
I understand where your mother's fear of faded colors comes from. Used to be that many darkly colored items were not "color fast" in the old days and the colors would run/fade and also discolor light fabric clothing.
When I went off to college in the mid-1970s, the washers in my dorm were commercial Maytag coin-operated machines with only two settings: Hot wash or Warm wash. There was no selector for rinse temp so I assume it was either warm or cold. There machines were clearly built before the 1970s energy crisis!!
There was often a wait of an hour or more on each machine, so basically you could run only one load at once (you never could use two machines at once unless it was late at night). That meant everything had to go into one load, and the coolest wash temp I could select was "Warm". I doubt if those machines had auto temp control, so back then "Warm" may have been hotter than today's "Warm". Often I would find whites such as socks, underwear, towels turned light pink or blue from something dark blue (jeans) or red in the load. Dyes back then were not colorfast and this was common to happen.
The following year, I moved to a smaller dorm that had a better student-to-washer ratio, so often one could use two machines at once, plus these machines were newer and offered cold wash (which I rarely used). Also, there was a selector switch for rinse temp and I generally used Cold for rinse. If I had NEW dark colored items, I'd wash in cold the first time, then wash in warm, sorting dark colors into a separate load. This ended the bleeding of colors issues.
Today, it's rare to find a colored item that bleeds, other than perhaps Madras plaid fabric from India. If you aren't sure, then wash once in cold water. After that, if the fabric is suspect, wash separately with dark colors only, but no reason not to use warm. For towels, it's silly to use anything cooler than Warm, and as sudsman suggested, Hot may be better in terms of killing bacteria.
Due to a medical problem, I am susceptible to fungal (ringworm) infections on my right leg, and was advised to use towels only ONCE and then to wash them on Hot. That way I don't reinfect my leg by using a towel more than once. This means a set of eight Costco bath towels are each washed on Hot once a week, and after two years (100 washes) they begin to look and feel a little worn and rough. So I have to replace bath towels every three years or so for a whopping $56. Big deal, at least it keeps the ringworm infections at bay.
I'd suggest you do at least your own laundry, including towels you use. You could use hot for towels, bed linen, socks, and underwear, and warm for everything else. Another reason your mother may be a cold water wash freak is that her old toploaders used roughly three times as much hot water per load when selecting "hot". So using cold water in the TL days really saved on energy costs. The savings for using cold in a front loader are diminished because you are already using a lot less water to begin with.
Adding to what others have said about detergents: you need to use an HE detergent. Regular detergent doesn't work well in a FL. Tide HE powder is affordable, about $12 for the 80 load box which lasts WAY longer than 80 loads in a FL. Use just 3 tablespoons (45 ml or a 1 1/2 coffee measure scoops) per full load in an FL of your family's size. Good luck!!
