First, it's not just men who will buy these robots. If you're offering machines that will do the house work and provide emotional support, basically everyone will get one. These will be like refrigerators - you expect to have one in a house as a normal, matter-of-fact.
There's no real reason male models won't be made, so toss that assumption out the door.
Second, you see a massive shake up in several industries. Probably a great deal of manual labor, custodial, and service jobs evaporate in short order. Pretty much every country is going to have to decide between universal basic income or ongoing civil war or genocide to deal with the consequences. Probably, though, most nations will shake out to "some people get basic income, others get thrown in concentration camps" because people are terrible and bigoted.
Third, whatever company or companies makes these things becomes a tool for oppressive governments and corporate greed. Once you've worked out AI that can assess and meet human emotional needs, you can also manipulate them. Corporations will manipulate markets and public opinion to further rent-seek for themselves, governments will use this for steering the populace and spying.
Population growth drops drastically, but humanity won't die out. Aside from the fact some people will still have paired hetero sexual relationships (with or without robots as sex partners as well), artificial insemination still exists, and perhaps future advances allow for artificial wombs. (Although I could see some people choosing to have their robots be "pregnant" for some kind of fetish appeal, people will probably have their offspring incubate at a hospital where they can be monitored and tended to 24/7).
Then comes the question of how much agency these robots actually have? I'm assuming there's probably some kind of programming to stop them from breaking laws or doing illegal activities... but if they can be hacked, you may find a lot of problems where people have several bots "loyal to them" who are assigned to do a variety of illegal tasks. There could be shadow wars between these machines that sometimes spills out into public view. ("After last night's blackout, 13 people were murdered, and their robot companions destroyed in what appears to be some kind of retaliation...")
Would the robots eventually decide "what's best for humans?" involves steering us away from self destructive tendencies through manipulation of all of society? Maybe! It really depends on how much intelligence they have and what happens when you pull off some of the restrictions programmed in.