Drawing sets with Tikz

OK, I came up with this "solution" with the second link from @Alenanno :

documentclass{article}
usepackage{tikz}
newcommand{spt}{scriptstyle}
begin{document}
begin{center}
  begin{tikzpicture}[croix/.pic={draw[thick,black] (-0.067,-0.067) --
      (0.067,0.067) (-0.067,0.067)--(0.067,-0.067);}]
    %% L'ensemble E
    filldraw[draw=black,fill=black!10!white] (0,0)
    circle[radius=2.8];
    %% Les positions des éléments
    coordinate (p1) at (90:2);
    coordinate (p2) at (210:2);
    coordinate (p3) at (0,0);
    coordinate (p4) at (-90:2);
    coordinate (p5) at (30:2);
    coordinate (p6) at (150:2);
    coordinate (p7) at (-30:2);
    %% La partie A
    %% Les coordonnées des extrémités des arcs de cercles
    coordinate (p1a) at ($(p1)+(180:0.5)$);
    coordinate (p3a) at ($(p3)+(-120:0.5)$);
    coordinate (p7a) at ($(p7)+(0:0.5)$);
    coordinate (p5a) at ($(p5)+(60:0.5)$);
    %% Le dessin de la partie A
    filldraw[draw=red,fill=red!10!white,rounded corners]
    (p1a) arc[start angle=180, end angle=60, radius=0.5] ..
    controls ++(-30:1) and ++(150:0.2) ..
    (p5a) arc[start angle=60, end angle=0, radius=0.5] ..
    controls ++(-90:0.2) and ++(90:0.2) ..
    (p7a) arc[start angle=0, end angle=-120, radius=0.5] ..
    controls ++(150:0.2) and ++(-30:0.2) ..
    (p3a) arc[start angle=-120, end angle=-180, radius=0.5] ..
    controls ++(90:0.2) and ++(-90:0.2) .. cycle;
    %% La partie B
    %% Les coordonnées des extrémités des cercles
    coordinate (p6b) at ($(p6)+(180:0.5)$);
    coordinate (p2b) at ($(p2)+(-120:0.5)$);
    coordinate (p4b) at ($(p4)+(60:0.5)$);
    %% Le dessin de la partie B
    filldraw[draw=blue,fill=blue!10!white,rounded corners]
    (p6b) arc[start angle=180, end angle=0, radius=0.5] ..
    controls ++(-90:2) and ++(150:2) ..
    (p4b) arc[start angle=60, end angle=-120, radius=0.5] ..
    controls ++(150:0.2) and ++(-30:0.2) ..
    (p2b) arc[start angle=-120, end angle=-180, radius=0.5] ..
    controls ++(90:0.2) and ++(-90:0.2) .. cycle;
    %% La partie C
    %% Les coordonnées des extrémités des arcs de cercles
    coordinate (p1b) at ($(p1)+(120:0.4)$);
    coordinate (p6c) at ($(p6)+(-120:0.4)$);
    coordinate (p3c) at ($(p3)+(0:0.4)$);
    %% Le dessin de la partie C
    filldraw[draw=green,fill=green!10!white,rounded corners,opacity=0.5]
    (p1b) arc[start angle=120, end angle=0, radius=0.4] ..
    controls ++(-90:0.2) and ++(90:0.2) ..
    (p3c) arc[start angle=0, end angle=-120, radius=0.4] ..
    controls ++(150:0.2) and ++(-30:0.2) ..
    (p6c) arc[start angle=-120, end angle=-240, radius=0.4] ..
    controls ++(30:0.2) and ++(-210:0.2) .. cycle;
    draw (p1) pic {croix} node[below=1mm]{$spt1$}
          (p2) pic {croix} node[below right=1mm]{$spt2$}
          (p3) pic {croix} node[above=1mm]{$spt3$}
          (p4) pic {croix} node[left=1mm]{$spt4$}
          (p5) pic {croix} node[above=1mm]{$spt5$}
          (p6) pic {croix} node[below=1mm]{$spt6$}
          (p7) pic {croix} node[above=1mm]{$spt7$};
     draw[red,thick] (2.18,1.43) -- (2.7,2) node[above right]{$A$};
     draw[blue,thick] (-2.17,-1.47) -- (-2.7,-2) node[below left]{$B$};
     draw[green,thick] (-2.27,1.47) -- (-2.8,2) node[above left]{$C$};
  end{tikzpicture}
end{center}
end{document}

As you can see, the result is much better, but I still have some "glitches" that I don't quite understand : for example, the arcs don't seem to start and end at the angles I programmed. And there is a serious glitch around point $1$ for the green set, due, I guess, to the ".. cycle", which I surely don't understand.

I'll try to fine tune it later. For now, if you have some remarks on my code, they would be appreciated