A lot of this depends on wealth.
While population density is high, about 4 times the current global maximum, there's no reason for it to be that much of a problem. If the apartments are well built and give families enough basics, if there's work, food and healthcare, it could just be a well regulated city. Dhaka has a population density of around 115,000/sqm. It's not massively high rise, just consistently 6-7 story buildings still set in space.
By preventing people from driving you've significantly reduced the air pollution, by putting everything within an easy walking distance you've probably increased everybody's level of daily exercise.
What this comes down to is whether the towers were built for this density or whether you've got 3 extended families living in a space designed for 2 people and a baby.
For the size and situation you've given there's no reason not to build a single solid development with ground level but overbuilt service roads. The lower levels for service and commercial activities and the higher levels for residential. Parks, gardens and the "street level" for walking on are actually a couple of stories above ground level, over the roofs of the commercial levels, there's no reason for the average person to even see a motorised vehicle in their daily lives.
Hospitals and healthcare
You're apparently suggesting a single hospital per city, 1 for 5-6million population. The equivalent value in the UK is 3.5/100,000 population (not all have all services). That's equivalent to 14/sqm on your map, a little overkill but certainly a 1/sqm is not unreasonable. That's not including GP surgeries, where you'll need a GP for every 4000 people or so. Centralising your healthcare is impossible and gives you no redundancy, you'll need to spread high quality facilities widely across your city. You can have a central hospital for all the real specialist services, but anything that people need day to day, GP surgeries, pharmacies, kidney dialysis, physiotherapy, geriatric care etc. etc. should be distributed.
Coughs and colds, seasonal flu etc. could possibly run rampant, but this sort of thing still depends on the general health of the population. If the water supply is good, the plumbing is good, and the people are looked after, that should be the worst of your problems. The less you centralise the slower diseases will spread. If everyone goes to the same place for their healthcare then a single infectious disease could hit the whole population in short order.
As per hospitals, distributed so children have the shortest possible journey to school for the duration of their compulsory education.
These can be centralised, they're more about area coverage than population. You'll need some system of constituency representation to be able to handle the population but that's administrative, but there's no reason they shouldn't be able work out of a single building. As per waste collection etc.
Again these need to be decentralised, as much for control of population as anything, you don't want too many people in one place if things get a bit rocky. You want nice controllable population elements so if one sector goes bad it doesn't spread so rapidly across the whole city.
Population density has made plumbing critical infrastructure and hence a government service. It can't be risked to the private sector.
It doesn't have to be a dystopia
but let's make it a dystopia anyway.
So far we've decentralised to a limited extent. Now we need to isolate people. One of the great failings with modern megacities is their excessive centralisation. People commute for hours to get to a job that barely pays the rent and transport costs. Welcome to the new utopia, no more commuting, everything you need is in your home tower. Your job, shops, gym and healthcare are all no more than a few stories from your home.
Excuse me sir, can you fill in a short survey on why you need to leave your home tower today? Is some facility lacking?
Well that's how it starts. Luxury gated estates are all the rage these days, people feel safe if other people can't get in. Gates work both ways though.
Halt, do you have a transit permit?
It won't take long to change from one to the other. Excessive movement of people puts excess load on the system, it means the logistics become complicated, the government won't know where to send resources causing waste in a system that must be efficient. The system must know where everyone is, people should stay where they're put.