Note that there are huge limitations. Windows software tends to write to video drivers, which then write to video cards. Windows software like TeamViewer can intercept the commands to the video drivers,and do fancy remote access stuff.
DOS software tended to use BIOS calls, or direct video writes. Direct video writes were said to be superior, as they were faster. So some software used them. But direct video writes do not offer much opportunity for remote access programs, or other programs like multitaskers, do be able to capture video well.
Video was a major problem where a lot of programs had troubles. To a lesser extent, keyboard access could also have such troubles.
Doorway was designed for serial ports (including modems, which used serial ports or acted similar to serial ports), not TCP over IP over Ethernet.
I do think that there were other solutions. They might have been commercial, rather than Shareware. They probably also all suffered the same limitations, for the same reasons.
Other programs may include Gateway2, PC Anywhere, and Carbon Copy. Some or all of those may have been commercial (with no demo/shareware/free version available). At the time that DOS was used more widely, I got the impression that these alternatives did not tend to work substantially better/different regarding the limited ability to work with DOS video and keyboard interactions.
In a nutshell, such programs didn't work well until newer operating systems became more widely used. You can try to use such programs, but expect that there may be challenges, including many cases where the software just does not work with the software that you want to use. Expect the system may hang up, be unresponsive, etc. That is just the reality of how far things had evolved in that time era. OS/2 and Windows 95 were vast improvements in these areas. (And, of course, Win98 was better than Win95 in at least some ways.)