Unfortunately, yes, it does.
Engineering isn't simply about technical ability (and what I read of Musk at Twitter did nothing to reassure me about that either), it's also about ethics and how you affect the perception of the engineering profession as a whole. In fact engineering competence is only one of seven attributes the Institute of Civil Engineers identifies for professional level membership - my father was CEng MICE*, so it's the one I have greatest familiarity with, but the other engineering institutions will have similar.
Amongst the subheadings in those attributes:
Commercial Ability
- Demonstrate sound judgement on statutory, contractual and commercial issues in relation to own area of responsibility
Confronting applicable legislation in multiple legal jurisdictions and refusing to comply doesn't present a responsible image for engineering or good management practise in general. See also the restrictions placed on his leadership of Tesla by the US SEC based on his irresponsible behaviour on Twitter affecting its share price.
Health and Safety:
- Demonstrate a sound knowledge of legislation, hazards and safe systems of work
- Demonstrate leadership by promoting good practice and improvements in health, safety and welfare
It's extremely questionable whether Musk's leadership at Tesla meets those requirements, particularly his claims around autonomous control and the actual reality. And see again sound judgement on statutory issues.
Interpersonal Skills and Development
- Proactively manage diversity and inclusion
Getting into a public spat with one of your employees over their reasonable adjustments for disability, and actively mocking them for them, then blaming others, really doesn't speak for any sort of understanding of diversity or the appropriate behaviour around it. Similarly sacking workers for speaking out about sexual harassment at Space X.
Ethics
- Understand the ethical issues that may arise in their role and carry out their responsibilities in an ethical manner
- Identify the limits of their personal knowledge and skills
Sacking 80% of the online safety team at Twitter and leaving child abuse safety tools turned off for months don't speak to a deep understanding of ethical behaviour. Nor do apparent sackings just for disagreeing with him - remember, this is precisely the sort of management behaviour that's caused the issues at Boeing.
And that, IMO, adds up to Musk being both a bad engineer at the individual level and bad for the image of engineering as a whole.
* Equivalent to the US PE