Is there a "UAP phenomenon" worth Studing?
First, some general perspectives:
For this response, I will take the perspective that what we mean as UAP phenomenon is really aliens or their technology near or on Earth. It might also include previously undocumented natural phenomenon that we have not yet found. I exclude distant life in other parts of the galaxy or beyond. I also exclude the uninteresting "unidentified" commonplace stuff.
It is not for you or I to decide what should be scientifically studied. Scientists will do this themselves. If something is interesting, then someone will study it. I don't buy the argument that scientists are afraid to do a study because of the stigma potentially attached to the area of study. It is true that scientists need funding and it may be difficult to get funding if there isn't a reasonable study design and the potential for interesting results. My opinion (which seems to be shared by many contributors here) is that so far there hasn't been a great deal of stuff to study and therefore no scientist has stepped up to the plate to study it. In adjacent fields of "is there life beyond Earth?", scientists have been studying this for decades. They don't seem to be bothered by stigma.
The scientific process requires that a hypothesis (at least one) is identified along with the null hypothesis. Once this is done, appropriate studies, tests, observations can be designed to reject the null hypothesis. If the null hypothesis is rejected at a sufficient (statistically speaking) level, then the scientist knows that they may be on to something and can devise further tests or revise the original hypothesis as needed. Fundamentally, if the hypothesis isn't falsifiable, then we aren't doing science. We can disprove a hypothesis, but it is exceedingly difficult to prove a hypothesis (most philosophers would say impossible).
As I've stated in in a previous thread on this topic, it is important to determine that there is something to study before spending effort on studying it. Besides the obvious issue of wasting time, effort, and money if there isn't something to study, we get into the "tooth fairy" issue. I believe the late Harriet Hall coined the term "Tooth Fairy Science". She said (and I am paraphrasing from memory, my apologies if I miss it a bit) that you could devise experiments about the tooth fairy such as analyzing the amount of money left by the tooth fairy over time or other factors that affect the amount of money left, but the results would be meaningless since you didn't start by ensuring the tooth fairy was a real phenomenon.
It is common when in the early stages of scientific discovery to conduct survey like efforts as part of developing hypothesis. The hypothesis being explored (possibly unstated) is something to the effect of "is there something here to study?" These surveys can be dedicated surveys or they can be collections of observations made over time by many people (not necessarily scientists). I would argue that UAP science is in the survey stage at best. Since there may in fact be nothing to study, it is prudent to minimize investment in time and money at this stage. Most (all?) of the data we have now on the subject of UAP is observational. Interestingly, we have a temporal set of observations that we can extract some information. Just two quick personal takeaways of these observations are: 1) they seem to be very tied to the gestalt of the time in history that they were made (i.e. we've moved from demons and angels, to nuts and bolts saucers, to orbs, to "plasmas"); 2) Even though our technology to observe and record observations has dramatically improved over the history of UAP phenomenon, we have not seen an improvement in the clarity of the observations. These taken together suggest that either the phenomenon is wholly artificial (e.g. Low Information Zone or social) or the real phenomenon is so subtle that we have just failed to capture evidence of it.
Now, what should we do? (again, "we" don't get to decide)
Without a doubt, there is fertile ground to study the sociological aspects of the UAP phenomenon. There is obviously something going on here, we have data, we could design experiments to collect more data, there are scientists that are probably interested in this. I don't know, but some of this has probably already been done. Studies of why people think what they think, how conspiracies arise, etc.
Another area that might be worth studying more is all around the neurological aspects of UAP perception. Why do people "see" UAPs. Lots of this ground has been covered both in general and specifically around UAPs.
On the topic of actually studying UAP themselves, I doubt that anyone (wealthy individual or government) is going to pony up the resources to exhaustively survey space, air, and water for the purpose of identifying UAPs as aliens. Our observations and coverage of these domains will continue to increase naturally over time due to other efforts. These include militaries defending nations, astronomers studying all things astronomical, searches in the ocean for missing ships or aircraft or resources, etc. Just the improvement in camera technology in the possession of consumers will increase the surveillance of our oceans and skies. The consumer grade IR cameras today are far better than the military grade sensors when I was working in the defence industry in the 1980s. If there is really something to UAPs, and people are "seeing" them, then it will show up eventually. I have serious doubts that this will ever happen, but I am open to evidence.
Finally, if any of the stories of crashed craft or recovered aliens is true, then it will leak. Scientific interest would grow fantastically if real physical evidence that stood up to scientific scrutiny came to light. All we need to have happen is one bit of true evidence to come out. The fact that it hasn't is not encouraging. It would take a very grand conspiracy to keep this stuff secret.