Yes they did, with absolute (or near) certainty they bred in 1416. There seems to be little to no evidence of breeding away from this single event.
There is also at least a place-name in the Knepp area. Given general scarcity of bird observation in the 15. century, two surviving records should be taken as regular occurrence. It is improbable that two isolated records both made it through public memory towards today. This is how historical records (not only birds and wildlife) are interpreted.
Your suggestion is against laws of probability, and against accepted methods of history.
- What is the temporal "ideal" ecosystem in terms of rewilding, and how can this determined? 50 years/100 years/500 years/1000 years
This question is not very relevant to reintroductions of single species, especially ones which have rather broad habitat requirements, like many birds. They would generally live in the 21. century landscape. Elsewhere, it would be interesting, if academic.
There is also, I suppose, a false dichotomy either-or, and an impossible assertion that all conservationists could and should agree on only one vision and one project.
- Why not invest in ecosystem scale conservation projects (in UK Great Fen project or Somerset Levels for example) rather than singular species vanity projects
You keep repeating the false argument there is some fixed pot of conservation money, and funds for reintroduction could be freely switched to other possible projects. Which is usually not the case, certainly not in case of White Storks, for example.
In addition to what Telephoto Paul written, charismatic species, especially big birds and mammals have special benefit of raising public interest. One could see reintroduction as an alternative to e.g. building yet another visitor center, explanation tables or boardwalks aimed at general public. Nobody questions spending money on these.
- If species are going to be (re)introduced, ensure that a EIA has been undertaken showing no detrimental impact to other species
You are asking for impossible and unscientific. In an ecosystem, any animal by definition has negative impact on its food species, and a plant at least on other competing plants. And you seem to be suggesting, that it is possible e.g. for a rare insect-eating bird to eat only common insects and never rare ones.
Besides, in absence of said species, checking or proving impact or no impact is normally not realistic. This is normally not considered relevant in rare European species, where it is assumed that since they coexisted in the past, and currently coexist elsewhere, they would fit together.
To add something: there is a well known tactic of delaying events by never-ending asking for more assessment, evidence, study etc. It fits your argument on proper spending funds - should one spend money on actual reintroduction, or waste on endless EIAs and paperwork, which often have little evidence to work with?
I would argue that for many reintroductions, it is better spending funds to try reintroducing species in many areas, even if they would fail in some of them. It would also mimic the natural dispersal and colonization, where most of colonizing individuals fail.