I don't think they're doing this for money. I think they're doing it ostensibly for security reasons, with the idea being:
If the unsophisticated user performs a search with the OS, and the links get clickjacked by a cracked browser and the user ends up installing malware as a result, they're not going to blame Firefox, or the malware-writer. They're going to blame Microsoft, because they were just using the OS. What's more the vulnerability won't be something MS can fix because it will be the fault of 3rd party software. So to prevent that possibility MS want end-to-end control over the search results.
Having said that I think a lot of the recent bad will toward MS is self-inflicted. One easy thing they could do is allow the paid versions (Pro and up) of Windows to have an "expert" mode that lets the user configure their OS the way they want, including removing all the telemetry and making all unessential apps/tools entirely optional, including Edge of course.
I forget the name of it, but there's a fork one of the open source browsers that's distributed by neo nazi types. I stumbled onto it trying to find a copy of XP I could use in a VM to run some ancient software. I wouldn't at all be surprised if it had a backdoor.
This was more along the lines of my thinking too. A more reasonable explanation here is that they this gives them control over UI and security which from MS's perspective may make sense if people are likely to attribute any UI / security issues to MS. It could also be something to do with the analytics they're able to collect.
Also when you have a company the size of MS talking about a small decision like this as if the whole company is onboard and actively engaged in the decision making process is naive. It was most likely a decision made by a few people at MS. Not saying that excuses it, but often individuals in companies have their own misaligned incentives and make certain decisions for their own benefits.
Are they still a dominant OS though? I'm sure they are if you're only talking about desktop operating systems, but I think we'd want to include mobile operating systems now that many people solely use phones and tablets for their personal computing.
If the unsophisticated user performs a search with the OS, and the links get clickjacked by a cracked browser and the user ends up installing malware as a result, they're not going to blame Firefox, or the malware-writer. They're going to blame Microsoft, because they were just using the OS. What's more the vulnerability won't be something MS can fix because it will be the fault of 3rd party software. So to prevent that possibility MS want end-to-end control over the search results.
Having said that I think a lot of the recent bad will toward MS is self-inflicted. One easy thing they could do is allow the paid versions (Pro and up) of Windows to have an "expert" mode that lets the user configure their OS the way they want, including removing all the telemetry and making all unessential apps/tools entirely optional, including Edge of course.