I think there have been a few studies, but can't recall off the top of my head. It hasn't been very fashionable for a while.
But Pheasants are fed every day by the keeper, sometimes twice a day, and sometimes all through the year. There are also Pheasant feeder hoppers all over the place. This probably provides almost all of their food, as the point of it is to stop them wandering off, and keep them in the area of the shoot.
Pheasants are also 'seasonal', in that they go into the pens around July, and are relased around the end of August, and most are dead by New Year. So it's less than half the year, and when everything is most abundant (seeds etc). The vast amount of extra food provides by keepers is also used by other species. I know some woods where the keeper spreads out half a tonne of mixed seed every single day of summer and winter, and a bit less every day in spring. This is dribbled off the back of a trailer as he drives round the woods, and 5 minutes later the tracks are buzzing with tits, finches and nuthatches. I have often also seen finches, buntings and tits taking grain from the pheasant hoppers.
So on the whole I think Pheasant shooting is probably beneficial to birds, as wild birds are essentially 'weeds' that thrive off the Pheasant crop - they also want the cover and food that is provided in abundance. Raptors also get plenty of food from the corpses that they come across - dead or alive depending on who you believe! But I think it's fair to say probably a bit of both, although the expected return on shot Pheasants as a % of released birds is about 30-40%. So that's 60-70% that dies through roadkill, predation and accidents etc., which is a lot of food for Kites and Buzzards (up to 28 million dead Pheasants, not including the millions more of RLP).
But botanists and herpetologists probably have different views, as pheasants add nutrients that may make the soils eutrophic and so encourage thriving stands of nettles at the expense of e.g. primroses and violets (but good for the Blackcaps!). They'll also eat lizards and snakes, and anthing else they can get in their mouths (including small birds and mammals). But again, they are only around when everything else is in abundance in autumn, so they'll be eating the 'doomed surplus' of voles and snakes etc. They are not around in any numbers in spring, so not eating up the breeding populations of these things.
In answer to your other question, I think Pheasant shooting went through a boom in the 1990s, thanks to the City bankers and the Guy Ritchie & Madonna effect (they made it fashionable for celebs). So lots of small woods started putting in pens where it was not economical before, and having a few days' shooting, which added to the many large estates that have always been there. But that has now come to an end, and the market seems to ave contracted. The small woods where a farm manager would dabble in 4,000 Pheasants 10 years ago now don't bother, as the economy has shrunk.
Climate change is proposed as the biggest problem for the future, but it hasn't had much effect yet. There have been lots of studies suggesting a link, but its usually expansion and not species loss. We're seeing some species apparently expanding north, but they're not contracting from the south.
