They should be learning them both. All of this discussion of "why not this instead of that" is wrong, because the children should be learning it all.
There is not a known limit to the amount of things a human being can learn. Each and every one of us should be trying to learn more, all the time. The more skills you have, the better.
You could learn to both type and write cursive, but you could also learn to both type and make balloon animals. Cursive writing in the U.S. doesn’t serve much of a purpose academically. I learned cursive as a child and have since forgotten because the moment I left elementary school there was never any reason to use it. Only cursive I use is to sign my name.
But does touch typing really require teaching? I learned to touch type just by typing. Gained my muscle memory naturally during regular usage.
I will say I hated being told to use home row in school, think that was only in one class though. Luckily the teacher only ever walked around every now and then. My hands were too big for home row placement.
This was slightly less than 2 decades ago, so computers were only in the computer lab and a few computer classes. This was mid aughts.
Touch typing does in fact require teaching. In the UK alone, only 20% of people can touch type fluently (https://touchtypeit.co.uk/just-how-many-people-can-touch-type). In the US, I saw a stat somewhere that was even lower at 10%, but I can't find it again. I wouldn't be surprised if that's accurate though.