If this were the case, I would expect to see the most cost-effective cases saturated (leading to reduced marginal cost-effectiveness), but with tremendous improvement reducing child mortality globally. But that apparently hasn’t happened, when looking at the income for those charities.

Is it really justified to say that this hasn't happened? After all, after you saturate all the most cost-effective interventions, all the previously-second-most cost-effective interventions become the most cost effective ones. There's always something that is most cost effective and unfunded.

Child mortality has in fact dropped sharply in the last few decades, and I believe at least some of that is due to international aid work and health improvements. An example I hold in my head (that is years old now, so possibly stale) is that vaccine distribution is extremely cheap as a way to save lives, and the only reason it's not an EA cause area is that it's already funded by existing institutions, and the only remaining things to fund in vaccine access are the cases that are very difficult to reach, or more indirect things like improving take-up among populations that have access.

(To be clear I'm not saying this to let billionaires off the hook; I agree that in general many / most of them are failing their moral obligations. I'm not sure how much of the above work was funded by philanthropists rather than governments or grassroots movements. But I think it's relevant context nonetheless.)

There is a post here that looks into this question. (Although almost 10 years ago).

"There are some good reasons for why large donors would want to not give too much money to a charity at once:

1 - Avoiding excessive reserves: Because of the opportunity costs (other charities could use money productively sooner), it is undesirable to have a charity having excessive reserves. Ideally, they would be promised a steady stream of funding if they meet specific targets over many years in order for them to be able to plan ahead.

2 - Risk diversification: Funds should be distributed to several high impact organisations in order to diversify the risk of one of them not performing well.

3 - Incentivizing others to join the cause area:
   a - Countries: By restricting funding to a particular country, one incentivizes the country to invest in very effective health interventions themselves and use their (often very limited) domestic resources to close the funding gap between donations and the full cost of delivering effective health interventions. Poorer, low-income countries (such as Ethiopia) are less able to do this than low-to-middle income countries (such as India).
   b - Charities: By restricting funding to charities, they’re being kept on their toes, so that they do not rely on a particular foundation or big grant giver exclusively and apply for other grants. For instance, in the past, the Gates foundation has heavily funded the Schistosomiasis Control Initiative. However, Gates later discontinued SCI’s funding not because of too little effectiveness, but because, since their effectiveness had been established, other funders would more readily fund them.
   c - Other donors: By restricting funding to particular charities, other donors are incentivized to also invest in the effective charities. For instance, the Against Malaria foundation has a broader appeal to small private donors than more high-expected-value interventions. Thus, even though theoretically, the Gates foundation, which is the largest private foundation in the world with an endowment of US$42.9 billion[4], could buy every person in Africa a bednet every two years (population of Africa (1 Billion) * Cost of Bednet (5 Dollars) = 5 Billion dollars) that would rapidly deplete their limited resources and then they could not spend their money on other very effective causes. They might reason that (small) more risk-averse donors (who want to be certain that their money will have an impact) will close the funding gap of very effective and established interventions and that they can instead spend more money on riskier, high expected value areas.

4 - Technological Innovation: New technological innovations—such as a very effective malaria vaccine—might be discovered, and these might be more cost-effective.

5 - High risk, high reward project:
For these reasons, one needs to come up with short-term room for more funding estimates.

Crucially, many room-for-more-funding numbers do not include scale-up and seem to be an estimate of how much money the charities can use to fund projects in the short-term (i.e. in the next couple of years). In other words, these estimates do not take into account potential mid-term scale-up of their interventions into other countries or into more projects (see for instance[5],[6]). For instance, Givewell reports that AMF’s room for more funding in the next year is $25 Million, which translates to roughly 5 million bednets. Because AMF has reported to be constricted in terms of funding in the short-term, they had not been actively reaching out to other countries to discuss new net programs in the beginning of 2015[7]. AMF also has more leverage with regards to national malaria control programs agreeing to implement their strict and rigid monitoring and evaluation process, when they agree to give governments millions as opposed to thousands of bednets[8].

Because these room for more funding estimates do not include scale up, the Gates foundation cannot simply close the funding gap for bednets."

I think there are a few different sub-questions here, with different answers.

If your question is, "Why doesn't the Gates Foundation fill the entire funding gap for GiveWell's top charities", I think there are a number of reasons why a value-aligned, careful, thoughtful, rational, and well-funded donor might nonetheless choose not to fill the entire gap. Open Philanthropy has written quite a lot about their thoughts here, and most of that would apply to any other funder. @DavidNash's answer seems directed at this sort of question.

If your question is, "Why does Larry Ellison give to the IDF instead of AMF", or "Why does Carl Icahn give to NYC stuff instead of the GFI"... I think the answer is just that billionaires are just regular people, like you and me, in a lot of ways, and in particular that the things that make one a billionaire philanthropist do not really guarantee that one will be an effective philanthropist. Billionaires have the resources to get all the advice they want, but most philanthropy advice is still basically "follow your passion". Most donors don't even say they care about effectiveness, and I doubt the picture is much different with billionaires.