What you descibe sounds most like the adapter is doing the best it can for what it is.
There are "sync" settings in (only) some VGA monitors that could compensate completly for a sloppy conversion.
The digital to analog converters and VGA monitors can also generally run best at specific resolutions, or aspect ratios. As you discovered they can still "work" at many resolutions, just a bit off.
HDMI output itself is very malable via many of the display drivers own software. It is possible you could tweak resolution (and the aspect of it) The display software, and or the monitor to get an acceptable picture. It is common for HDMI output to have to be adjusted in display properties for the GPU software to compensate for Overscan.
Are they only capable of a few resolutions NO. Is 1366x768 a weird resolution that a cheap converter might leave some blank space or offset picture? YES. Is it possible that even the best converter would have some offset or issue? YES. If it was going to an old-school CRT high-end monitor could most of it be compensated for? Probably.
Find everything to adjust, and adjust everything, hours later you might get it closer.