Why are lute-like instruments found in so many cultures?

I am not an ethnomusicologist so I can only give a speculative, rather than an authoritative answer, but by examining the particular qualities of the instruments you describe we can come up with a plausible explanation for why those qualities are seen in such a variety of instruments.

Stringed instruments (chordophones) are one of the major categories that almost all traditional instruments fall into; the others are membranophones (beating a membrane, as in a drum), aerophones (a vibrating column of air, as in a flute), and idiophones (the whole instrument itself is struck, e.g. a bell or cymbal). Given that most world instruments are members of these four categories, it is unsurprising that any single category will have examples all over the world.

Parallel strings — having decided on the use of strings to make musical notes, and further decided that we want more than one string to be available simultaneously, mounting those strings approximately parallel is the most logical and straightforward choice. It minimises the distance that fingers must move to choose between strings, it allows the string ends to be mounted near one another in a similar fashion, and it requires the instrument to be strengthened primarily in a single direction. If the strings were mounted at right-angles or in a radial fashion, the instrument would be much more complicated to build and play.

Resonating body — this is a necessity if the instrument is intended to be loud enough to be heard by an audience or in a social setting. If you've ever played the strings on an unplugged electric guitar, you can hear how quiet a string sounds without the resonating body.

Neck — this does vary between instruments; as you correctly point out, harps and zithers do not have a neck. However note that these instruments are either non-portable (the Chinese Ghuzeng, concert harps), or use a separate string for each note (harps and lyres). If you want to be able to use your finger to change the pitch of a string, as in a guitar or violin, you need something behind the strings to press against, and this board does not need to be much wider than the area occupied by the strings themselves. Having a narrow neck with five or six strings allows you to fret or stop the strings with a single hand, while using the other hand to pluck or strum them, making for a very convenient and portable instrument.

My (theoretical) conclusion, therefore, is that "lute-like" qualities derive naturally from the practical concerns involved in making a portable and audible musical instrument based on the principle of vibrating strings, and do not in themselves require the existence of some common ancestor.