It reads to me that the series of decisions you have made as a couple have meant you have arrived at a situation where there isn't a lot of give to your life. It's a pretty impossible situation and if it continues as it is I don't think you are going to make much progress in the situation.
The change to your employment, whilst didn't cause your partner to need to work nights, means that now there is a significant barrier for your partner changing jobs.
The night shifts are having a negative impact on his wellbeing. Definitely pursue a review on the sleep apnea, suggest restarting the antidepressant but also research SAD and maybe look into getting a SAD lamp and tips and resources for improving SAD in shift workers. I'm bipolar and this time of year does a number on my mood. The days and days without sunlight really kick my mood.
As he is working nights, struggling with the lifestyle changes this brings, as well as, and I say this as kindly as possible, your strong desire to be present and always there mum means that his role of dad is squeezed into 12-20h a week. Some of the language you use alongside your critique of his interactions couples with a person that is susceptible to depression I can see how your partner might then struggle with the being a present parent. Defaulting to "my" son and saying he will "ruin" bedtime. I have a poor sleeper, I get it, but can you also see that how the negative language perpetuates the perception that what your partner does isn't going to be right and I can see how it would then be harder to initiate play that you are going to be judged on. Your son will be aware of the dynamic too and you won't be equal, you will be the person that answers 97% of his needs and daddy is the person he tolerates the day you aren't there.
I struggled with playing with my daughter due to severe pnd and one of the things I found helpful wasn't the big activities and the dedicated play time but doing more of the day to day activities and learning to be a bit more playful in those brief moments. It meant that my daughter saw me more playfully and we could bring those bits of play into activities. I had some therapy with the infant parenting service and they did some recordings about aspects of parenting and we then looked at clips and photos showing those aspects of parenting. It was really helpful and allowed me to slowly get out of my head. I really struggled with letting loose and being childish as it felt so false.
My eldest is 5 and a poor sleeper, I have a 5 month old and I think you are quite naive in thinking that you are going to be able to carry on as you are. We had to start trying to get my daughter to sleep for my partner around the age of 2 as I found myself in a & e with my mum after she fell down a flight of stairs fracturing her back in two places. It was bed time and I needed to be in the hospital for my mum but also at home for my daughter. For two weeks I left the house every other night at bedtime so that dad could do bedtime. There were lots of tears the first few nights but she quickly accepted daddy at bedtime. I then did another two weeks every other day being " busy" at home and after that she has accepted both of us for bed time. It was liberating for me, knowing that if something happened I could be somewhere else and it strengthen my daughter and partners relationship. We have had to change her routine since our second arrived in September. She was still quite reliant on us for sleep but then when baby was 5 weeks my partner had a meeting in the US. His parents came up to help, in the first few weeks I had been able to feed the baby and get him to sleep and then do bedtime with my daughter but at 5 weeks we started with witching hour at my daughters bedtime. We had 4 days of my in-laws trying to do bedtime with my daughter but her wanting me, my son needing me too. We were playing pass the parcel with the baby and me trying to do as much of my daughters routine as possible. Then the day my partner got back I had a him with horrific jet lag having been awake for 32h, a screaming baby and a 5 yo that wanted mum to sit with her to go to sleep. I couldn't give anyone what they needed. The 5 yos bedtime routine had to change and ever since she has gone to bed on her own and dad has done most of them. We had hoped to change her bed time routine before baby arrived but I got so unwell in pregnancy I had 2 assessments for preterm labour, twice weekly blood tests and two hospital admissions before being rushed into hospital for induction at 36+6 weeks. I was too unwell to look after her during the day so had to prioritise her being ok with others during the day to worry about the night time
There's not much that can be changed whilst you are on maternity but afterwards could you adjust the balance a bit, would there be opportunities for you to do an extra shift and your partner do a day less? You might find that extra day makes the difference to his energy levels and resilience.
I'd look into things you can do for yourself once a week for half an hour. I think it does a person good to have time when they aren't parent/partner/child. I started an online pilates class. I'm not a gym goer and can't stand group classes but the online class just feels like me and the instructor and it helps me breathe properly for 45 minutes a week. I have previously gone running and swimming just a bit of time where I'm not having to be or do anything, just be me feeling the beat of the pavement or the flow of the water. I have got back into reading since having my daughter and when I was working I would sit and read on the train to work but now I'm at home I try to have a bath and read for half an hour.
In the long term you need to look into both of yours employments to work at what the right balance is for your family as a whole.
In the medium term you need to look at improvements to your partner and your health and wellbeings
In short term look at giving your partner more time to be default parent, step back a bit and let him do some figuring out for himself. Play doesn't have to be structure activities and for hours at a time look at the small moments too and there are some more immediate things regarding his health that want looking at sooner.