Diagrams have a “sans-serif” look-and-feel

I certainly do not precisely know how serif works, but if I look at the appearance of a \Delta then it seems that some line widths are larger than others. Analogously, the contours of Dynkin nodes might want to become a bit thicker towards the south east. The perhaps simplest way to achieve this in the given settings is to add a tiny black pseudo-shadow to each of them.

EDIT: Proposal after feedback.

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[full]{textcomp}

\usepackage{tikz}
\usetikzlibrary{calc}
\usetikzlibrary{cd,positioning,shapes}
\usetikzlibrary{decorations.pathmorphing}
\usetikzlibrary{decorations.markings}
\usetikzlibrary{backgrounds}
\begin{document}
Let $\Delta$ be a connected Dynkin diagram, and
let $\Delta^+$ be the extended (or affine)
Dynkin diagram associated with~$\Delta$.
Then $\Delta^+ = \Delta \sqcup \{\alpha_0\}$.
Below we depict the connected extended Dynkin diagrams,
in which $\alpha_0$ is depicted by a grey node
\tikz \node[draw,circle,inner sep=2pt,fill=gray] {};.
 \[
 \begin{tikzpicture}[
  Dynkin/.style={yscale=1.2,draw,circle,fill=white,minimum width=8pt,inner sep=0pt,
  append after command={\pgfextra{\begin{pgfonlayer}{background}
                \draw[yscale=1.2,fill=black] ([xshift=0.18pt,yshift=-0.15pt]\tikzlastnode) circle (4.2pt);
                \end{pgfonlayer}
            }}},
  every label/.append style={rectangle,font=\footnotesize,
   inner sep=1ex,text depth=1pt},
  decoration={markings},
  doubledynkin/.style={double distance=2pt,postaction=decorate},
  a0/.style={fill=gray},
  every shadow/.style={fill=black,opacity=1,shadow xshift=0.5pt,
  shadow yshift=-0.2pt},
  Dynkin line/.style={preaction={transform canvas={shift={(0.2pt,-0.2pt)}},draw,
  #1}},
  ]
  \node[draw=none] (A1text) [label=right:{\normalsize$A_1^+$:}] at (-.5,0) {};
  \node[Dynkin,a0] (A10) at (1,0) {};
  \node[Dynkin] (A11) at (2,0) {};
  \draw[double distance=2pt,
  decoration={
  mark=at position 0.95 with {\arrow{>}},
  mark=at position 0.36 with {\arrow{<}}
  },
  postaction=decorate] (A10) -- (A11);

  \begin{scope}[xshift=5cm]
   \node[draw=none] (Antext) [label=right:{\normalsize$A_n^+$ ($n \ge 2$):}] at (-1.5,0) {};
   \node[Dynkin,a0] (An0) at (3,1) {};
   \node[Dynkin] (An1) at (1,0) {};
   \node[Dynkin] (An2) at (2,0) {};
   \node[Dynkin] (Annm1) at (4,0) {};
   \node[Dynkin] (Ann) at (5,0) {};
   \begin{scope}[on background layer]
   \draw[Dynkin line] (Annm1.center) -- (Ann.center) -- (An0.center) -- (An1.center) -- (An2.center);
   \draw[Dynkin line=dashed,dashed] (An2.center) -- (Annm1.center);
   \end{scope}
  \end{scope}
 \end{tikzpicture}
\]
\end{document}

i also do not understand what is meant by the "more serif" experience. used font is "serif" ... for nodes i suspect, that you like to have ellipses (obtained by scaling, which influence also on font size) instead of circles:

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[full]{textcomp}

\usepackage{tikz}
\usetikzlibrary{arrows.meta,
                decorations.markings,
                positioning,
                shapes.geometric}

\begin{document}
Let $\Delta$ be a connected Dynkin diagram, and let $\Delta^+$ be the extended (or affine) Dynkin diagram associated with~$\Delta$. Then $\Delta^+ = \Delta \sqcup \{\alpha_0\}$. Below we depict the connected extended Dynkin diagrams, in which $\alpha_0$ is depicted by a grey node \tikz\node[ellipse,draw,semithick,fill=gray,inner xsep=2pt,inner ysep=3pt] {};.
 \[
    \begin{tikzpicture}[
    node distance = 10mm and 10mm,
every node/.style = {ellipse, draw, semithick, inner xsep=3pt, inner ysep=4pt},
every label/.append style = {label distance=1em, rectangle, draw=none},
        a0/.style = {fill=gray},
doubledynkin/.style={double distance=2pt,
                     decoration={markings,
                         mark=at position 0.9 with {\arrow[semithick]{Straight Barb[length=5pt]}},
                         mark=at position 0.1 with {\arrowreversed[semithick]{Straight Barb[length=5pt]}}
                                },
                     postaction={decorate},
                     }
                        ]
\node[label=left:{$A_1^+$:}]  (A10)   {};
\node[right=of A10]           (A11)   {};
\draw[doubledynkin]   (A10) -- (A11);
%
    \begin{scope}[xshift=5cm]
\node[label=left:{$A_n^+ (n\ge 2)$:}] (A1) {};
\node[right=of A1]          (A2) {};
\node[a0,above right=of A2]    (A3) {};
\node[below right=of A3]    (A4) {};
\node[right=of A4]          (A5) {};
%
\draw           (A2) -- (A1) -- (A3) -- (A5) -- (A4);
\draw[dashed]   (A2) -- (A4);
    \end{scope}
\end{tikzpicture}
\]
\end{document}

in above code i use positioning library for relative placement of nodes. with this you can with change of node distance simply change all distance between nodes. beside this, i simplify image code with removing all unnecessary nodes.

addendum for fun and exercise :-)

added fancy stuff as copy shadow, one emphasized (thicker) line between the most left and top node in "triangle" nodes formation, by moving styles definition in document preamble, overall code is slightly shorter ...

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[full]{textcomp}

\usepackage{tikz}
\usetikzlibrary{arrows.meta,
                decorations.markings,
                positioning,
                shadows,
                shapes.geometric}
\tikzset{Dynkin/.style =
{
every node/.style = {ellipse, draw, semithick, inner xsep=3pt, inner ysep=4pt,
                     fill=white,
                     copy shadow={fill=black,
                                  shadow xshift=0.6pt, shadow yshift=-0.2pt},
                    },
every label/.append style = {label distance=1ex, rectangle, draw=none,
                             every shadow/.style={opacity=0}
                    },
        a0/.style = {fill=gray!50},
doubledynkin/.style={double distance=2pt,
    decoration={markings,
     mark=at position 0.9 with {\arrow[semithick]{Straight Barb[length=5pt]}},
     mark=at position 0.1 with {\arrowreversed[semithick]{Straight Barb[length=5pt]}}
            },
    postaction={decorate},
                     }
}% end of Dynkin style
        }% end of tikzset

\begin{document}
Let $\Delta$ be a connected Dynkin diagram, and let $\Delta^+$ be the extended (or affine) Dynkin diagram associated with~$\Delta$. Then $\Delta^+ = \Delta \sqcup \{\alpha_0\}$. Below we depict the connected extended Dynkin diagrams, in which $\alpha_0$ is depicted by a grey node \tikz[Dynkin]\node[a0,scale=0.75] {};.
 \[
    \begin{tikzpicture}[Dynkin,
    node distance = 12mm and 12mm,
                        ]
\node[label=left:{$A_1^+$:}]  (A10)   {};
\node[right=of A10]           (A11)   {};
\draw[doubledynkin]   (A10) -- (A11);
%
    \begin{scope}[xshift=48mm]
\node[label=left:{$A_n^+ (n\ge 2)$:}] (A1) {};
\node[right=of A1]          (A2) {};
\node[a0,above right=of A2]    (A3) {};
\node[below right=of A3]    (A4) {};
\node[right=of A4]          (A5) {};
%
\draw[semithick]    (A1) -- (A2)
                    (A4) -- (A5) -- (A3);
\draw[very thick]   (A1) -- (A3);
\draw[semithick, dashed]       (A2) -- (A4);
    \end{scope}
\end{tikzpicture}
\]
\end{document}