It's true that increasing trail and rake both increase stability. But curiously, a bicycle can be made stable with both negative rake and negative trail.

An interesting take on this, with both mathematical modeling and real prototypes, can be found at [1]. Note also that gyroscopic stabilization is not necessary: in the prototypes, a counter-rotating extra wheel cancels out the angular momentum of the front wheel.

According to the authors, it's not yet even proven that a stable bicycle must turn towards a fall. Almost the only sure thing, so far, is that "at least one factor coupling lean to steer must be present". We know a lot of sufficient conditions for stability, but not what is necessary.

The conclusion of the paper: "As a rule, we have found that almost any self-stable bicycle can be made unstable by misadjusting only the trail, or only the front wheel gyro, or only the front-assembly center-of-mass position. Conversely, many unstable bicycles can be made stable by appropriately adjusting any one of these three design variables, sometimes in an unusual way. These results hint that the evolutionary, and generally incremental, process that has led to common present bicycle designs might not yet have explored potentially useful regions in design space."

EDIT: here's a not-paywalled version of the paper linked in the submitted article.

[1] PDF link: https://scholar.vt.edu/access/content/group/141943de-4222-46...