The world would not actually be improved by having 1000s of customer service reps genuinely authentically feel sorry. You're literally demanding real people to experience real negative emotions over some IT problem you have.
The exceptions are generally when people are scared, and sadly some people are scared all the time.
People require empathy and compassion; we need others to mirror our emotions and indeed to share them with us. We are social creatures and it is not normal, healthy or effective to experience (strong) emotions alone. Connecting emotionally with others is not a luxury or weakness, and certainly not "bad"; it's how humans naturally and essentially function.. Yes, it can be done badly and you don't need to be powerless - if your partner comes to you terrified about a cancer diagnosis, acting terrified yourself isn't helpful; but accepting their emotions, seeing them, and responding with genuine emotions appropriate to the situation is essential.
Many highly analytical people - to use a vague, undefined term - tend to think that anyone who comes to them with a problem must want their problem analyzed and solved - if you have a big hammer then all problems are nails, I suppose. Sometimes that is desired but certainly not always, and it can work against what people really need.
My last comment was intended to be read in that context too, not about interacting with the people you're close to.
I still think you can have empathy on support calls; I'd even say it's important for the customer to be satisfied. They may be panicked, frustrated, exhausted, etc. Ignoring people's emotion gets bad results; it feels rude.
Of course there are limits, especially time; long stories are inappropriate. Still, I've had many empathetic, brief conversations with strangers on trains (literally) and elsewhere.