Well, clearly the Oberth hull form can be set up to do just about anything you want it to - but can any given Oberth type do everything? Or are they all a set of specialists built on the same hull? The Grissom is clearly good at science, but evidently bad at combat. The Biko is being used for supply transport, with no evidence that she does anything else. The Cochrane is only ever seen as a personnel transport. (Yes, she reports heavy casualties in the Dominion War. That doesn't mean she was good at combat. It just means she got shot at. The Lusitania reported heavy casualties in 1917 and no-one is suggesting she was good at combat.) The Oberth herself was assigned to a Deep Space Exploration mission both when the Bozeman got caught in the time loop and during the events of TUC, officially 15 years later, suggesting that she is good at science and exploration. The Pegasus was a technology testbed, with nothing to say she ever did any other job. The Tsiolkovsky is only seen after doing science. The Vico is heavily modified for very specific science, so she was presumably doing something else before - but her tasking here is in line with the Pegasus and the many other science ships. The Yosemite is again doing science - though given the crew complement, may also have been testing technology.
In fact, the only time we see an Oberth do something that isn't either transport, science, or technology testing (arguably a variation on science) is the Bonestell, which gets destroyed at Wolf 359. But it's clearly established that Wolf 359 is a desperate last stand made up mostly of entirely unsuitable ships, so Bonestell being at that battle is not evidence that the ship was any good at combat. She was just close enough to die trying.
Obviously, much of this comes from the storytelling structure. Cochrane aside, we only ever see any of these ships doing a single mission. Who knows what they'd do next week? So there's no real evidence that Oberth-class ships are specialist ships that only perform one type of task. But there's equally no evidence they aren't. If you prefer a setting where every Federation ship is a multi-role generalist and the only real difference is "big ships are better at everything," go you! Still, my interpretation that most ships are specialists and only the relatively rare cruisers are maids of all work is an equally viable conclusion.
