SSE isn't really more complex than long polling. The only difference is the server don't close the connection immediately after sent the response. Instead, it wait for data again and send more response using the same stream.
One limitation of SSE compared to long polling (and WebSockets etc) is you can't efficiently send binary data such as cbor, protobuf, etc. Though if your long polling is chatty enough eventually the HTTP overhead will kill your efficiency too.
Long polling is more amenable than SSE for most HTTP tools out of the box, eg curl. The SSE message body is notably different from plain HTTP responses.
To the OP, you can still build APIs with long polling. They are uncommon because push patterns are difficult to design well, regardless of protocol (whether long-polling, SSE, websockets, etc).
Whiteboarding a push API is a good exercise. There is a lot of nuance that gets overlooked in discussions whenever these patterns come up.