If you want to be a software engineer, study software engineering, not some other engineering area. I think electrical engineering is far less useful to people if they are going to be general software engineers than being a cs major. You will spend a lot of time learning things that won't matter to software engineering. Of course if you were going tl be employed as an electrical engineer, but if you only were trained as a software engineer, you wouldn't be able to be proficient either. Study the field you want to work in!
In my experience (a very long way from SV) a significant proportion of excellent programmers did not do SE degree but had degrees in some other area. My analysis suggests that critical thinking, ability to comprehend requirements and do good design are critical skills and can be developed in many related fields.
