Welcome to OSM, and thanks for improving the map in your area! You’ve happened to touch on a topic that lacks consensus in the community, as evidenced by a recent 55-post thread on this forum just a couple of weeks ago! To spare you the reading assignment, a slight majority of the participants there actually favor the mapping approach shown in the screenshot, with the “stubs” separated and mapped as sidewalks, though clearly a significant number of mappers prefer other approaches, like including them as part of the crossing way or using a separate footway= value.
There are some reasons typically put forward in favor of mapping crossings the way shown in the screenshot: you’re right that the stubs tend to have similar attributes with adjacent ways, but usually the attributes (surface, smoothness, etc) are the same as the adjoining sidewalks, not the crossings. And as you surmised, mapping these ways precisely allows one to put the curb/kerb nodes in the right place topologically and physically. But it’s also true that it adds complexity, and whether it ‘feels’ right to have these conceptually as part of the sidewalk or the crossing seems to differ from person to person. I’m of the opinion that the screen-shotted micro-mapping style is “the most correct”, but to be honest, often times as a first pass I omit the sidewalk stubs, hoping I or someone else will come back later and add more detail. Obviously you can read more (a lot more) about each side’s pros and cons in the linked thread.
Whatever style you prefer, I would recommend not putting the kerb/tactile paving nodes directly on the intersection between the crossing and the main sidewalk. That’s because kerb nodes are often used in wheelchair routing, which can avoid things like raised curbs. Putting it directly on the main sidewalk way makes it seem like the whole corner is impassable, whereas really if one turned the corner without crossing the road, they wouldn’t encounter a curb at all.