What could make a male-only human reproduction system exist, which makes women obsolete, expendable, and also gives men full control over their paternity?
The basic mammalian incubation organs are just not there so they will need to go back to external incubation, which means either an artificial test tube, or going back to laying an egg. The first case is relatively easy but requires a very advanced technology.
To do this biologically, without added technology, the men will have to both create eggs, lay them externally in some sort of nest, and fertilize eggs since there is no way to combine the gametes inside a male body. It will be similar to the way most fish fertilize eggs.
1. Generation of eggs
This requires a genetic modification of the chromosomes so that essentially all children are true hermaphrodites with ovotestis. Sorry, you can't avoid needing an egg. But this way, one human can make both egg and sperm, so you will not have discrete male/female genders based on anatomy. The down side is that a person can self-fertilize, and this will be very bad for genetic diversity. Normally this condition will have very abnormal humans, so the genetics has to be fine tuned so a "stable" hermaphroditic condition can be reached.
2. Laying eggs
Likely both eggs and sperm will be deposited in a nest area, which has a very high chance of self-fertilization. Eggs and sperm therefore need to have some way of being separated when deposited. This can probably be done with a low technology screen, or maybe even a drug that neutralizes either the egg or the sperm, but not both. Someone has to decide whether they are going to be the "mother" or "father" of the baby in advance, and take the correct drug. Theoretically, the outcome won't be very different either way, so it's a simple clinical decision with almost no impact.
3. The nest
You will need to simulate the natural environment of a human womb for about 9 months. That means a very nutrient-rich amniotic fluid-filled egg that maintains a constant 98.6° F. This is hard to do and will require constant feeding by the prospective parents. I imagine the incubation pool will be sterile saline water with some sort of artificial amniotic sac, that soaks in a pool of circulating water to keep warm. It needs to be treated like a bird's nest with constant attention and maintenance.
These are the fundamental components needed to eliminate the need for women. It will take a massive amount of genetic engineering either way, because you still have to come up with eggs that males just can't make as we are today.
Very rare cases have babies successfully-ish develop in woman's body parts other than the Uterus. https://en.wikipedia.org/wiki/Ectopic_pregnancy#Live_birth
Fiction had it happen to a man via artificial means.
Science says it's probably possible, but the risks can't be justified. https://en.wikipedia.org/wiki/Male_pregnancy#Humans
As for conception, I suppose it's possible via very artificial means to fuse the nucleuses of two sperm cells (at least one being X) and putting the resulting full-genome nucleus into a male steam cell strong-armed into becoming a zygote.
Maybe birth is still random for the women
When a child js born, the DNA and gender is randomized. Since women just procreate with each other, what if they can still create male children? Men would not have to worry. With this, there could be some adoption, surrogacy, lesbian couples procreating, etc. Social norms and equality may take longer in some places, or be completely different, but males will still keep on existing, just not having much sex for reproduction.
The laboratory solution
This answer looks at what biotechnological challenges you would have to solve if you had two men and want to create a baby who is the genetic offspring of both of them. Assumption: You have a lot of very smart microbiologists and bioengineers, and you can provide them with nearly unlimited resources for research.
Women have two X chromosomes. Men have an X chromosome and a Y chromosome. So it would be genetically possible for two men to have a child by taking the X chromosome of one and combining it with the Y chromosome of the other. It would even be possible to create a completely normal girl by taking the X chromosomes of two men. So if you don't want to reintroduce women into your world and return to the old-fashioned way of making babies, then you might need to come up with a reason why this is not possible or not desirable.
Step 1: Acquire an egg cell and replace its genome
Now this is the hard part. In order to create a new human, you need a human egg cell, and replace its DNA with the DNA from the sperm cell of a man. The easy solution would be to simply take a natural egg cell from a woman which is still available, for example because it was preserved from when women still were around. But any frozen egg cells will run out sooner or later, so you need a more sustainable solution to acquire new ones.
Perhaps it would be possible to convince a stem cell from a man to turn into an egg cell. This would require a lot of research, but could be a plausible approach.
Step 2: Fertilize it
Now you need to take your egg cell with the DNA of the first man and fertilize it with the sperm cell of the second man. This is the easy part. In-vitro fertilization is a solved problem.
Step 3: Incubate it
Artificial wombs are another science fiction technology which appears possible in theory, but still needs to overcome several problems in practice. The technology does not exist yet, but some of its parts are in active development. It would be plausible for it to exist in the near future.