Scanlines depend on the TV/monitor you are using. The thick heavy scanline look came from tri-sync arcade and PC monitors. When displaying 31khz (VGA) about 480 lines, scanlines are tiny, however when those monitors display 15khz, only about 240 lines are used, hence there is empty space between each line resulting in those thick scanlines.
However older mono (15khz) and dual (15/24khz) displays do not display the image that way, they still do have scanlines, but they are only as thick as the space between pixels, and the spaces are the same distance in both the horizonal and vertical lines, meaning you get a tiny grid as opposed to the above lines.
What you prefer is in the eye of the beholder. I prefer the older original grid look, but with the popularity of candy cabs in home use, the super thick lines are what a lot of people have become used to.
Since scanlines are the space between pixels, generally the larger the display the wider they will become. For a more accurate measurement the dot pitch rating is the amount of space between pixels.
Sony TVs work a bit differently, their Trinitron tech displays differently then standard types. Iirc, tritrons use octogonal pixels as opposed to the standard squares. For this reason even the standard SD Wegas tend to have thick heavy scanlines.
The Sony HD-CRTs use special tech and upscales everything. Hence 240p/480i content is upscaled and you lose the scanlines completely, they also will have issues with things like 3D LCD shutters and light-guns because of this.
In the end though you really need to find a look that suits you. For years lots of people were quite happy with their Sony PVMs, but then the thief fudoh proclaimed that Sony BVMs were the best picture ever, so everyone started dumping their perfectly good PVMs for BVMs. Don't fall into that trap, just get a decent RGB display that fits your needs and be happy with it.
For myself I prefer to use a dual-sync when I can, but the X68000 requires a tri-sync so I have both. Not counting weirdo devices that need specialty monitors (FM Towns).