You both need to put in an effort and stop making excuses...
Why don't you foresee yourself having a date night anytime soon? You said you live near family, why can't they babysit? You have a caregiver during work hours, can they care for your children after hours? Can you both take a day/afternoon off and have a date while the children are being looked after? If family and their current caregiver cannot offer occasional care, then hire a babysitter. There ARE options.
Why can't you put the kids to bed earlier? Young children do not need to be up until 9PM, and depending on the wake-up time it's not even healthy. I understand wanting to spend time with the kids and as a family, and you should, but you also need to spend time together as a couple. On a related note, you really need to think about whether the income from your wife's job is worth it.
I know what it's like to be busy as hell, have a bunch of little kids, deal with pregnancy complications, months of postpartum healing, months/years of rejection and sexlessness, resentment, cranky babies, babies who refuse a bottle, sleep deprivation, co-sleeping, etc. BTDT - I have six kids (6 months, 1, 2, 6, 12, and 14). Our frequency has been all over the spectrum but currently we're doing it at least once a day. I wouldn't say that's normal but where there's a will, there's a way.
Start having a date once a week. Move the kids bedtime earlier and spend an hour (or more) together every night. Get up early with her to squeeze in some time. Try to get your kids to have an overlapping nap. Teach your kids to have quiet time in their room or another safe space. And you can be spontaneous even with an infant - to go with the example given, put them in a seat in the bathroom and hop in the bath/shower with your wife. I've done it many times. It's not as good as doing it alone, but it's certainly better than nothing.
If you haven't already, have a vulnerable talk with your wife about why you are so afraid to initiate, what it felt like to be rejected when you finally tried again, etc. She needs to know how her actions effect you, and no it's not to guilt trip her.