"Scientific Investigation" presumes that science can provide all the answers.
I disagree. Scientific investigation is a rational method of trying to find an answer to many questions, but there is no presumption that it can provide all the answers.
Sometimes it is not the right tool, just as the best toxicological analysis is no use if you are trying to find out what sort of knife was used to stab someone.
Scientific investigation relies on a falsifiable hypothesis, a sufficient sample of data, and results that can be independently reproduced. This approach is no use at all if you are trying to understand a single anomalous report. You have one small set of data, no opportunity to experiment, and no way of reproducing your results.
However, a rational approach to finding explanations would include applying scientific knowledge and other sources of experience to the problem. For those who strive to explain anomalous reports, a fair consideration of the available evidence, the significance of the absence of evidence, and the credibility of the witnesses are all vital.
If there is physical evidence, such as a photograph or a clump of supposed yeti hair, then forensic-style analysis is a valid way forward.
Having said all of that, I dislike the original question, "What stance
should Forteans take...?"
I have my preferred approach; you may have yours; and someone else may have a third way. It's a hobby for some, a superficial interest for others, and a passion for others. Who am I to say how someone else
should approach Forteana?
Although I use the word myself in this forum, one danger of the word
Fortean is that it may attract a formal definition, and that the field will then have self-appointed gate keepers: "You're not a real Fortean because you don't accept the central tenets of Forteanism."
There are no such central tenets. Forte was not a messiah, nor even a particularly naughty boy. We may or may not be his successors, but we are not his followers.