The short answer is yes, starting with version 2.0 and even more so with Warp, OS/2 was a viable alternative for daily DOS and Windows tasks, up to and including Windows 3.x.
Starting with OS/2 2.0, the first 32-bit version, OS/2 included very, very good support for DOS programs, including protected-mode and some measure of direct hardware access. This allowed it to run Windows 3 itself (with some modifications), and most real-mode and DPMI protected-mode DOS software.
It was thus compatible with most protected-mode DOS programs (an early demo involved running Doom in a window), most Windows 3 programs, and even Win32s-compatible 32-bit Windows programs (at least on OS/2 Warp 4). Odin, released in 1999 and continually in development since then, is supposed to provide more general Win32 binary translation, but I haven’t used it much. I don’t think Windows 95/98/NT compatibility was ever all that good in practice, and it certainly wasn’t when OS/2 was still relevant — the industry’s switch to Win32 was the nail in the coffin for OS/2.
Note that OS/2 does require more resources for basic use than DOS and Windows, so when it first came out, many potential users would have been frustrated if they’d tried it — it really needs at least 8MiB of RAM (which was still unusual in early 1992) and a decent amount of disk space. However on a system with enough resources to run it well, it really is very fast, and perhaps more efficient than DOS and Windows.
Some consider that this excellent compatibility contributed to OS/2’s demise: since it was so good at running DOS and Windows programs, there wasn’t much incentive for developers to provide OS/2-specific applications. There were a few, but no big killer app sufficient to carry the platform and allow it to compete with Windows. The requirement for Windows compatibility, which was always based on running a dynamically-modified version of Windows inside OS/2, meant that users needed either a Windows license, or a version of OS/2 with its own Windows license, which meant Microsoft made money off of OS/2 sales anyway (above and beyond any remaining licensing agreement from the joint IBM/Microsoft days)...
This episode of LGR Tech Tales gives a decent summary of the life and non-death of OS/2 (which lives on as ArcaOS).