I think they died well before responsive became a concern. Their heyday was really the time before not just CSS but even tables, at which time they were the only way to have links positioned in space rather than in flowing text. I don't think anyone liked imagemaps - it's just that there was no alternative.
But their limitations were massive. Chiefly, in a period when dialup internet was becoming more common, an image map meant that before a user could interact with your website, they had to wait for a very slow download of a very large image that contained mostly text. You couldn't interact with them any more - no ability to select the text in them, or perhaps the reader had a different color/font preference? Tough luck.
When tables came out, they began to face competition, but they weren't defeated. Aside from table-first designs that mixed images and text, you could mostly do without imagemaps by cutting the image up, eliminating table borders/cellpadding/cellspacing and making careful use of colspan and rowspan, with a part of the image in each cell. This had most of the same user-side problems as imagemaps, but was easier for many people to maintain (I'm not sure quite why: I assume some people have trouble with thinking in terms of x/y coordinates and regions).
CSS thoroughly destroyed them. Everything an image map could do, tables+CSS and eventually CSS could do better. This was still ages before responsive became a concern, at a time when webpage maintainers would assume you used an 800x600 or 1024x768 screen with a maximized window and just ignore everyone who didn't match that.
Nowadays, any time an image map might look like the solution, the actual solution is SVG, which gives you everything an image map had but is moreover correct.
