From what I can tell, the situation in Perth is probably going to last a few years, and each year, a new crop of bright young engineers is going to graduate into a pretty dim situation. However, being in the east, I don't know what it is really like there.
On the other hand, as you say, for some, it is really difficult to move. This may be especially true for you - you may need to move first and be followed later by your SO, so that she can switch jobs...
Many years ago, the job of my SO moved to Canberra from Melbourne, where he had lived and had always planned to live. He moved but was very unhappy in Canberra - he hated it. He drove the 8 hours to Melbourne and back every second weekend, starting on Friday at lunchtime, and (because he could work a nine day fortnight due to flex time) driving back each Monday. Looking back on it, I think he never gave Canberra a chance, because his "life" was in Melbourne. That lasted for nine years.
Then I was seconded to Canberra, and moved here. Moved house, and finally had a home together again. He likes Canberra. He wants to live here for the rest of his life.
You will probably love Sydney (I don't - but I've never lived there) if you give it a chance. Personally, if I were you I would take the chance because you are currently underemployed and hating it, have limited prospects where you are and have been offered a job in a city where other prospects will be easier to find. However, if I were you, I would also have a six month reality check and live apart from your SO for that time.
Before I retired I managed a number of people, and often hired people - including people from Perth. Sometimes things worked out, and sometimes they didn't - and people who were in a completely new city without family and friends were more likely not to work out (they usually did work out). In my opinion, it would be better to have six months apart while you ensure that you can live in Sydney, and are happy with the new job. Also, as a manager, we often had people on probation for the first three months. If you do have a probation period, it would be better for your SO to work out a transfer after that is over and after you have decided you are not completely adrift in Sydney.