TL;DR: There is always some score-keeping happening in relationships, but to get the benefits of score-keeping you're going to depend hugely on how you keep the score and even more on how you communicate about it!
Plain score-keeping like you propose here probably won't work. Even if you both agree to this system initially, it will probably just escalate into other arguments. Just to illustrate: If your husband is spending quite some time getting a fussy baby to sleep, and you can do the dishes, a load of laundry, and wash the windows in that time, you will have three points against his one, but you'll both have spent the same amount of time. How many changed diapers is a cleaned room worth? And what if you do a thing that's not on the list? One of you will just use the list to start more arguments, in my experience.
I've heard the 'do not keep score' argument before. Mostly when my mom would tell us kids to do things, and I'd complain it wasn't fair. My arguments were usually that I'd already done X and thus a fair share of Y was actually unfair, or they were based on expectations that my much younger youngest brother (we differ 10 years) would have to take an equal share instead of a share that suited his age at the time.
For mature, adult relationships, the power dynamics are different. There is no longer a mom dealing out the chores, instead you have to come up with a fair way of sharing on your own. And this can be hard! I've discussed my parents way of dividing work with other people, and some would call it unfair, while others would call it fair enough. Just to show that many people will have many different views on what is fair. Keeping score like you propose will likely go badly, because... How many changed diapers is a load of washed, dried and put away dishes worth? Your partner and you are going to have different opinions on that, and that's why plain score-keeping doesn't work.
But there's always some sort of score keeping already happening in relationships, as they are seldom truly altruistic and rely on some kind or form of reciprocity to work. Reciprocity is defined differently by different disciplines within the social sciences and humanities, as can be seen in this overview of wiki pages on the topic. But all those forms have one thing in common: A human expectation to get something in return for something else.
In your system, you're expecting your husband to change just as many diapers as you do, and your husband will have the same expectation. Someone changes a diaper, and they expect to get out of changing a next diaper in return. This isn't necessary reciprocity though, as the 'return' can be something different from the original 'favor', and score-keeping like that will never work. This article mostly matches my experience and puts it in a good wording:
if you focus on what your partner is not doing, you will only see these limitations and not look for his or her positive contributions that can increase relationship satisfaction.
While I don't have a partner, I do have a few brothers that I've also had some trouble with in the past. Summarized, when my parents weren't home they would skip chores that they usually did, and not help out with the work that mom and dad usually did for us. Keeping a list of who did what never helped, because I would indeed focus on all the things I had done and all the things they hadn't done.
What helped me to improve the problem with my brothers was switching 'score-keeping' system I used and communicating differently. I suggest you may want to change yours too, and perhaps your husbands one as well.
Instead of keeping score of all the things my brothers hadn't done, or all the things I had done instead of them, or how I was doing way more things than them, I switched to noting the things they had done.
For example, these days my youngest brother has a quite physical job, so of course he doesn't feel like doing the very physical household chores when he comes home! But, he starts late so it's no problem to ask him to e.g. clear out the dishwasher in the morning and put in a load of laundry so I can hang it when he's working. He can also do grocery shopping, or take the car through the car wash. In return, I make sure I do the vacuum cleaning when he's at work, and hang/fold that laundry. I cook, and we share the cleaning of the kitchen afterwards, with him taking out the trash and wiping the counters, and me loading the dishwasher and handwashing any items that can't go in there. So, instead of tracking all the times he never hangs the laundry, I've started tracking the moments that he does help out with other things.
And if there's still stuff I think is missing, I switched the way I communicate about it. Instead of my communication noting something isn't done yet, I make it a request to do so in the future while I'm doing something else. An example can be found in the article I linked earlier:
or example, saying, "You never clean the kitchen!" is a lot different from saying, "Can you please help me with getting some things done around the house this weekend?" Being mindful of your urges to use negative words and tone, and then alternatively being kind, really makes relationships feel more positively connected.
I'm doing much the same with my brother now. Instead of telling him he hasn't cleaned the kitchen, I make remarks like "I'm going to fold the laundry now, would you mind wiping down the kitchen in the meantime?". These work well, because it shows I'm also doing something! They work a lot better than telling him he didn't clean the kitchen while I'm playing videogames, or asking him to do so while I am watching Netflix. It helps 'keeping the score' visibly, because after all we're both doing some chores at the same time and can't argue about 'I already did this so now you have to do that'.
So, my conclusion at this point is that the proposed way of score-keeping will most likely end with one of you tracking things that the other hasn't done, and starting arguments about the value of items on the list, and that as such it's not good to keep scores that way.
Instead, you have a better chance of resolving these conflicts if both of you stop noting the things that aren't done and focus on the things that are done, while asking each other to do help out if more help is needed. If you want to keep score a bit more visibly, ask your partner to do one thing while you do another, or let the partner pick one of two chores and do the other yourself.