Indeed, in many sources, the int10h subfunction AH=0 to set video mode returns without any meaningful status.
One way, suggested by C&T BIOS programming guide, is to verify if the mode is set via Read Current Video State function, int10h with AH=0x0F.
It will return the current active mode among other things.
In more detail, the MDA and CGA cards have no BIOS of their own, so IBM 5150 motherbord BIOS has built-in support for them. And in fact you have to set with motherboard DIP switches which adapter you have (MDA or CGA, and if CGA, which default mode you want, 40x25 or 80x25).
The point is, if your switches is set to MDA, it gets stored into Equipment Flag bits, and when setting a new mode with Int 10h, the Equipment Flags are checked for display type.
If the Flags indicate your swiches are set to MDA, then, no matter what mode number you request, the mode 7 parameters will be set to the MDA card. So you can't set any other mode.
The IBM manuals say if you have both MDA and CGA, you still have to set the switches to MDA, so you can't select other modes through BIOS.
If you don't have an MDA, but only CGA card, BIOS initializes either mode 1 or 3 based on if the switches are set to 40x25 or 80x25. If you have MDA or both based on switches, video init code tries to select mode 0, but the mode set function always forces the mode to 7 in case of MDA present.
This leads to an interesting use case, if you have CGA and no MDA, what happens if you select mode 7? I think the BIOS tries to write the MDA mode register settings to the CGA card, as I see nothing preventing it.
Which means, a game cannot know through BIOS if it can set mode 7, so maybe best to check what is the current mode before trying to set any mode, and it seems having dual MDA/CGA card combo means the BIOS must either be tricked to CGA only mode via the Equpment Flags, or the game must detect and control the CGA card directly without the BIOS.
Having a later card such as EGA or VGA made this easier as you can just set a mode and it works, or detect the presence of EGA or VGA directly.