In the last method, you need to draw a different tangent line. Well, it's just a line. But it's tangent to the larger circles, and tangent at the point where a smaller circle touches the larger circle. Maybe I can add some notes to your file..... I'll attach it shortly....
Edit
Well, I didn't have to do much to show you. Since you have everything already set up so precisely, I just had to draw a 45° line. And since you have snapping set up as well, it snapped perfectly tangent to one of the larger circles. I colored it red for clarity.
If you move that line out to the smaller circle, the place where it intersects with the smaller circle is the same place where you want the other lines tangent to the smaller circles to start. Since I'm not familiar with this gasket, I'm not sure what decides where those lines need to start. So I'm not sure exactly how you would place the red line across the smaller circle. But you must have had some idea about that, unless you just kind of "eyeballed it". But anyway, with the red line on top of the circle, Path menu > Cut Path will break the circle in the proper places. Then you just delete the unwanted part.
Or, since you're interested to learn, there's another way, which gives the same result, with one less step. So much with Inkscape is a matter of preference, and learning what works best for you. Anyway, notice the blue rectangle I made. With the rectangle on top, select both the rectangle and circle, and do Path menu > Intersection. You'll have the same result, and it saves the step of deleting the unwanted part.
One more note. This might be unnecessary. But it seems, from what I can tell (I don't have any kind of cutter) that some cutter machines raise up the cutting head or laser at the end of a path. You probably would get the same result either way. But to make sure it cuts all in one go, without starting and stopping, you probably want to join the end nodes of all the paths, so that you end up with a single, closed path.
To do that, switch to the Node tool. Select 2 adjacent paths, like one tangent line and the remnant of the circle. Presumably, the end nodes will be right on top of each other, which means it's really hard to select whichever node is on the bottom. So to select them both, you have to drag a selection box around them, just around the 2 end nodes which are on top of each other. Then click Join Selected Nodes button on the control bar (or use the key shortcut, whatever it is)(I tend not to use key shortcuts, if there's a button for it, but that's just me). Then continue around the whole thing, until you have a single path.
Actually, it might be possible to join them together in one step, instead of joining one segment at a time. Or at least join half at a time. Select all the path segments. Then while holding the Shift key, drag a selection box around each set of overlapping nodes. (It looks like there must be at least 16 segments, maybe 24 segments? So there would be one less number of sets of overlapping end nodes.) Then Join Selected Nodes button. It can be a little tricky sometimes. Once in a while, I find what I thought was an end node, actually there are 2 nodes there, and that keeps it from joining properly. It might join the wrong segments together, or it just doesn't join them. But once you get some practice, you'll learn how to sort out the problems, and be able to take little shortcuts like that.
Sorry for all the text. I just enjoy helping new users learn and sometimes I get carried away
