Dear Dean, you wrote:

"There might be approaches which could address such issues but most scientists are simply not interested. Many also feel that expressing an interest in this could hurt their scientific career.

I'd be interested in other's thoughts about this."

I know two serious attempts at scientific investigation of the matter of reincarnation/rebirth. First is of American (but Canadian-born) Professor of psychiatry Ian Stevenson, who analyzed many cases of remembrance of previous lifes and studied uncanny cases of inborn injuries or birthmarks on those areas of the bodies of supposedly reincarnated person, which were damaged due to their violent death in their "previous life/lives" (accident, murder etc.). Stevenson, being a scientist, tried to apply rigorous scientific methodology in his research, but, as the critics pointed out later, he still could not avoid bias in his research, so most of it is considered to be flawed.

A very recent attempt is that of ven. Bhikkhu Anālayo, a key figure in current scholarly research into early Buddhism. In his book "Rebirth in Early Buddhism and Current Research" (2018), which I can highly recommend as an introduction to that area of inquiry, he takes up the research of Stevenson, provides a necessary theoretical background for the study of reincarnation and adds new empirical dimension to it: a philological one (!). He argues that a Sri Lankan boy, who at a very early age "remembered" his previous life as famous ancient Buddhist commentator Buddhaghosa, "remembered" the portions of the ancient Buddhist canon that are unique, having analyzed the tapes of the recitation of the canon by the said boy. 

Regarding your initial question: as Prof. Rupert Gethin answered above: the Buddhist tradition (at least the Southern one) holds, that rebirth is (or may seem) random from the point of view of an unenlightened person (i.e. including our modern science and philosophy), but strictly determined from the point of view of the Buddha. I may add here that the Buddha says that one should not think about the workings of karma(n), because it will drive them insane (because of complexity of the matter).

Kind regards,

Gleb Sharygin