Yeah, that shift-and-add algorithm is sometimes used on microcontrollers, either in software if there's no hardware multiplier, or in hardware if you want the bare minimum in acceleration at a tiny cost in area.

Adds are not really considered negligible; the article is just sloppy. (Some shifts might be negligible in some models because a fixed shift requires no logic gates.) The cost of the adds in Karatsuba is significant both theoretically and in practice, and determines the cutoff where Karatsuba is useful. But the exponent in O(n^(log_2 3)) is dominated by the recursive multiplications; the adds only affect the leading constant hidden in the O().

> If we're disregarding shifts and adds as completing in negligible time

When considering multiplication algorithms the parameter N is the number of digits of the two numbers. In that model adds do not complete in negligible time, and instead take O(N) time.

Russian Peasant Multiplication (https://www.embeddedrelated.com/showarticle/760.php)