More than you think. They are also actively seeking ways to make that leverage effect more people.
They are defining web standards. They control chrome and chromium. So all of the alternative browsers that aren't safari and Firefox are using Google's web engine. Even Firefox and safari are beholden to Google as they fund both these web browsers through their default search deals.
Google after many failed messaging apps has taken on RCS messaging. They provide most of the supporting infrastructure through their Jibe servers. They don't allow anyone but themselves and Samsung to make an RCS app on android. They also had a campaign to pressure apple to use RCS. It's likely apple's RCS will be following Google's Jibe service closely, as they've already said their will work with Google on this. Google successfully got most RCS messages going through their servers, with apple on board with RCS itll see most SMS messages defaulting to RCS and most of those going through Google.
They also have deep hooks into education market with their OneDrive/Google docs products and Chromebooks.
Most privacy focused android alternatives recommend Google hardware.
It was very likely a designers decision. It forces the use the use case they wanted; wireless mice should be used wirelessly. I would bet they fought marketing and management to get this on the final product.
Marketing would want the mouse they can advertise as being useable with and wireless. Female ports are easier to mount and manufacture with they have depth to set the socket. So a plug on the front is much cheaper and easier to manufacture.
The fact the charging cable doesn’t get used in motion means it will last longer and you wouldn’t have people useing fraying cables on the front of their mouse.