The most obvious way to verify if birds are stable or not in at least gliding flight would be to make the exact rigid model of the bird (including weight distribution) and launch it into flight. The simplest approach that even early aircraft inventors may have tried is to use a frozen real bird.
Are there any references about anybody trying such experiments? Pure logical or mathematical analysis is outside the topic of this question.
The closest I know of are several attempts to build models of pterosaurs.
Since the head with its long beak is ahead of their center of gravity, the configuration is statically unstable in yaw and needs continuous control corrections. On the other hand, this instability provides very quick responses. This YouTube movie contains footage of a Pterodactyl model made by Paul MacCready in flight. Note that Paul had attached a conventional tail for the first flights to statically stabilize the model in pitch and yaw. Later in the flights this tail was removed. The small skin area between the legs was most likely not sufficient for pitch stabilization, and small changes in wing sweep were used to stabilize the animals in pitch, just like modern birds still do.
Another branch, the Rhamphorhynchoidea, had a tail which provided additional lateral and longitudinal control. Palaeobiologists are still arguing, however, whether the tip of the tail was oriented horizontally or vertically in flight.
A number of experiments and papers by Robert Hoey (from the early 1990s on) have used models of flat-winged soaring birds such as ravens.
One of his papers is referenced in this broad review: Lentink & Biewener 2010
One of his earlier papers is copied here: Hoey, 1992, American Institute of Aeronautics & Astronautics
The topics include some discussion of low static stability of the model and the lack of vertical surfaces in real birds, as well as unknowns such as wing profile.
I don't know about birds but researchers have been working for a long time to replicate insect flight. This article talks about how 3D printing has made this much easier. It has a video of a working replica hovering. They call it an ornithopter. Technically "ornitho-" means "bird," but I guess entomopter didn't have the right ring to it.
I remember reading something back in the 80's about spy agencies trying to develop a "fly on the wall" cyber-insect that could get in through air ducts or other tight spaces.
