The speed of it is quick, and it also gives a good chance for wrapping things up (can we get this packed, thank-yous, goodbyes if it's a bigger group, etc).
Of course, it may also help that we have chip-and-pin and tap pretty universally, so we rarely hand our card over to anyone - having the server take the card is incredibly strange for me when I'm down in the US.
That said, I'm with you; I much prefer to have my card handled just once and within sight. It is faster, safer, and, to my mind, simply more convenient.
And that is considered "cheap"! By contrast, in the European Union there's a cap of 0.3% on credit card interchange rates and 0.2% on debit. Of course that's just the raw interchange fee; a small merchant has to pay more than that to its card processor, but there are all-inclusive, Square-like offers for debit card processing fees below 1% for small merchants, and larger ones manage to keep both debit and credit processing cost way below 1%.
So the "comfort" of being able to hand over your credit card to someone who takes it out of view and possibly does unauthorized stuff with it increases prices of all the stuff you buy by roughly 2-3%.
In the EU, you ask for the bill, they bring you the bill and the credit card machine, and you are charged right there at the table, where you have to sign (because you have an US Credit Card after all)
Actually it's kind of funny when you visit remote places in Europe, and the proprietors are confused because this is the first time they've rang up a US Credit card, and the machine spits out a second receipt asking them to validate the signature.
In the American model, payment is processed away from the customer and the bill is left for tip-adding after the server has left, to save the sensitibilities of cheap customers who tip badly.
Also, I thought tipping itself isn't necessary, as many european countries have social programs to offset low wages.