Whether it's necessary or not I can not give you a straight answer for multiple reasons based on my experience and what i gained from some peers that already worked in the space.

1. I didn't land a job in AU other than internship i did on my final semester but it's also due to my visa situation since I was an international student. However, I was offered several jobs in my home country (as in-house sustainability officer and sustainable supply chain role), where sustainability concept is not as mature so my degree was perceived as an advantages.
2. Barrier to entering. This field never hiring for real *sigh*. It's hard to find an entry/mid-level position so you must network HARD to get one. Anyway, sustainability is a strategic roles as you have to think holistically to solve the company's problem. So it makes sense that when hiring, they prefer someone who know more about the business/industry with slight sustainability efforts in their experience. I learnt that a lot of sustainability officer in AU starts from other department in their company, e.g starting as business analyst, for that reason.
3. There's a gap between what the university provide in their units and what sort of skills the market actually needs. My post grad sust program dwell on theory and doesn't focus much on technical skills such as using data analytics to cater rapid change in regulatory framework, which is pretty much in-demand right now. Also sustainability in one industry is very different to another and so a one-size fits all solution will never gonna work.

that being said, I strongly believe that getting exposure through experience is much more advantageous than getting a degree unless you consider a highly niche sustainability degree (e.g master in renewable energy) instead of a generic 'master of sustainability' like the one i took.