Though, you're still limited to string representations that way. If I want to examine a really complex datatype, I'll sometimes write a gdb command to dump some kind of GraphViz output and run it through dot. Or come up with a json representation and explore it using jq or ad-hoc python filters.
Looking at your ideal tool checklist - I think it would be possible to build this on top of gdb. I wrote a prototype a while ago that would launch a HTTP server from the gdb prompt, and then you could control it via websockets. Never took it past recreating the prompt with an <input> tag, but someone who was better at front-end work could conceivably do all sorts of nifty data visualizations with it.