It's clearly hard, but there are tools for doing exploratory visualization of high-dim data. GGobi http://ggobi.org/ and all the ones that arrange points but try to get local neighborhoods correct (t-sne, umap, et al.).

Yeah, but still "scary" because you have to be really careful to not fool yourself and pay attention even with those algorithms. For example, a good demonstration with tsne https://distill.pub/2016/misread-tsne/?hl=cs