Why these illusions, which is what they must be according to your worldview, are so frequent

That’s the right question. When I discard the obvious claims of reincarnations of famous people whose biographies are well known, or trivial unprovable cases, etc. and stay only with the cases that really could suggest reincarnation (however, I know of none — but that does not mean they don’t exist), plus if I consider my own experiences and what we already know about how brain, perception, and our senses work, I would say, the answer is something like this:

• our senses collect enormous amounts of information, which also includes disturbances/noise in that information (i.e. happenings that are normally out of our sensible spectrum)
• most of that data gets filtered out by our awareness before it reaches our “waking” perception, but is nevertheless “recorded/stored”
• brain, among many other things, can create simulations (what we call creative thinking, but also dreams or intuition fall into this category), i.e. it creates connections between seemingly unrelated chunks of stored data
• because thinking is pretty energetically demanding, brain (not wasting resources) does not run many simulations, but only those that are most essential and most likely to bring the whole organism into equilibrium (i.e. state of lowest excitement — in more spiritual words, peace)
• the above point is also supported by the fact that most (maybe even all) instances of that process happen in response to (existential) problem solving (even if we only aimlessly ruminate, there are subconscious motivations always going on) — but this “creative thinking” certainly always happens in altered states of awareness (daydreaming, regular dreams, meditation, ecstatic trips, etc.)
• now, the perception (i.e. that which we actually are conscious of) gets “compiled” (with the disturbances/noise rendered as patterns that we already know — hence six-handed gods with strange faces, etc.)
• the whole package is filtered by our awareness and finally cognized

In short — we construct the “reincarnation flashback” from information that is available here and now (plus memory). When you consider the causal nature of the universe (i.e. that all actions alter the flow of causal streams, or tao), all information reaching back to the beginning of creation is perceivable or at least inferable from the current state (in all places simultaneously), and we have access to it simply with our five senses (sometimes even less than five) and our brain.

And you can explain many so called “psychic” phenomena by this.

Ah, and the word “personal” was superfluous, indeed ;)