On the issue of waking up in the night, I'm willing to accept I might be wrong, but my thinking is: if my other half is on maternity, at least she can catch-up on sleep during the day (i.e., when the baby naps), whereas I have to work all day. I have said to her that I'm happy to do it on weekends when I'm not at work, but I think she expects more.
I don't know why anyone hasn't really picked you up on this but yeah, you're wrong.
You had acknowledge that while the baby's awake it can be hard to get things done right? Even putting on a wash can be a bit of a problem, so when is a good time to do these things? Yeah... while the baby's sleeping. If it stresses her out not being able to do these things then she's not got a chance of sleeping.
I mean, just think about it rationally for when she does to get a nap:
1) That baby will sleep whenever it wants
2) The second that baby's head hits the pillow you need to run to bed and start TRYING to get to sleep
3) The baby stirs, makes a noise, coughs - your heart misses a beat just as you think you might have been getting to sleep
4) You get 10 minutes of peace and you finally drift off
5) The baby wakes just when you don't need it and suddenly you're feel worse than you were before.
Again, this is a discussion better had with your partner, but let us know her face when you tell her she can "at least catch up on sleep while the baby sleeps" - because... just... no.
Remember that you'll never see the worst of it. When you're on paternity I assume she was home as well? At the very least it's a second pair of eyes to give eachother a mental break while the other keeps an eye on the little one. You'll likely never experience the stress of being home alone with all that responsibility on your shoulders so make sure you understand exactly what she expects from you.
One last time, nothing is better than keeping open communication with your partner. Don't take to heart everything you read online, or from others, talk to your partner.
Like I say, google "mental load" and here's another take on it below:
https://www.bellybelly.com.au/baby-sleep/things-mothers-really-do-when-baby-sleeps/
The whole notion of daytime napping is impossible for mothers also who have pre-school aged children, who need mamma’s time when the baby sleeps. It completely voids any opportunities for catching any sleep, unless she has an extra pair of hands helping her out while her partner works.
At the end of the day, while it may help to solve one problem if you sleep when the baby sleeps (if baby stays asleep long enough to make it worth it), it poses another big problem. It means that mum is stuck in a cycle of being awake when the baby is awake, and being asleep when the baby is asleep. Why is that an issue? It means she gets no time for herself, or for the important things that need doing – things that wont do themselves. Sure, baby sleep problems don’t last forever – but we all need a break. If you had to do your day job from the minute you woke up, and then had to sleep at your desk, you’d go a little crazy too.
So before you volunteer some baby sleep advice, be prepared to listen, empathise and support, rather than give an opinion that she didn’t ask for. Its best to wait to be asked for advice when it comes to babies. Trust me, many mothers do get the idea that sleeping when the baby sleeps means they may get a little more sleep (if they don’t get stuck lying there in bed, thrashing over their to-do list). Ask her what you can do to help instead, and she’ll love you forever.