What is the modern alternative to sieving soup recipes?
The modern way is still to sieve.

I actually did this a couple of days ago. I made a sauce out of boiled onions, capiscums, chilli peppers, canned plum tomatoes, carrots, garlic, ginger and spices. This was then whizzed in a blender and sieved through a fine metal sieve to remove the pulp, seeds, large fibrous pieces that wern't blended.

It makes for a smoother sauce.

The fine metal sieves are fairly cheap and easy to obtain. There are also finer sieves made from plastic or even muslin you can also use.

The technical term seems to be chinois.

A food mouli: