Wow, that was easy. Dispatching a 'change' event works for all three JS-rich elements: To-Community dropdown (after setting its .selected), Icon dropdown, and Tags string input (after setting its .value).
Here's how to do it for Tags:

const CLS_INPUT_TAGS = "autocomplete-input";
elTags = document.getElementsByClassName(CLS_INPUT_TAGS)[0];
elTags.value = "tag1, tag2";
elTags.dispatchEvent(new Event('change'));