Totally possible, just a horrible idea.
React uses a Virtual DOM to determine what attributes/elements need to be updated when changes occur. This way the cost of DOM updates per render call are minimized to only what is needed, or avoided altogether.
When you use
dangerouslySetInnerHTML, you simply replace the DOM tree every render call which can be very unperformant depending on the size and complexity of the tree. As you stated, you also lose JS events since SSR only outputs static HTML and things like
onClick are removed.