DOS-era games (and some early Windows games) had very limited controller support due to various technical limitations; the most advanced controllers at the time were not universally supported, and even they had fewer buttons and axes than a modern controller. Your physical controller doesn't really matter since the limitations are inherent to the games themselves.
You can use the
DOSBox Mapper (Ctrl+F1) to work around in-game controller limitations but don't expect the seamless experience you get with games that are built for modern controllers from the start.
OldFatGuy: Stargunner (one of the free games here at gog) offers built in native support for the SNES controller (download the game and go into settings and see for yourself) and I happen to have a USB SNES Controller, have been playing Zelda classic with it.
Nope, won't do a damn thing on Stargunner.
DOSBox is not an exact replica of a DOS-era machine. Games don't see your real controller, instead seeing whatever model has been set in the DOSBox configuration. The common models DOSBox supports should be sufficient for almost all games.
Exotic controllers didn't necessarily work that well under real DOS either; Stargunner's v1.1 patch dropped support for Sega and SNES controllers due to issues with the adapter.
OldFatGuy: My number one complaint about games here at gog. They get them to run, but they ain't exactly running like they did originally, because using Redneck Rampage as an example, me and my son spent probably hundreds of hours playing that, with a controller, that was as easy as plugging in and selecting it under settings. Good luck with that today with the same game and even the same damn gamepad/controller.
DOSBox's joystick support is mostly very compatible but for some games you might need to adjust the joystick model and timing in the DOSBox configuration file; a more appropriate model will ensure the biggest number of buttons and axes are available (assuming the game supports it) and the timing setting resolves various problems such as the controller not calibrating correctly in-game or moving by itself.
There isn't really much GOG (or the DOSBox team) can do about this since DOS controller support is subject to various technical limitations of the era.