I looked at the texinfo.tex file and it appears that a short table of contents just redefines the macros for sections and subsections to null.
Here then is a diff:
diff -u texinfo-hacked.tex{~,}
--- texinfo-hacked.tex~ 2019-10-07 16:44:56.000000000 -0400
+++ texinfo-hacked.tex 2019-11-29 12:41:46.105737894 -0500
@@ -6881,15 +6881,18 @@
\rm
\hyphenpenalty = 10000
\advance\baselineskip by 1pt % Open it up a little.
- \def\numsecentry##1##2##3##4{}
- \let\appsecentry = \numsecentry
- \let\unnsecentry = \numsecentry
- \let\numsubsecentry = \numsecentry
- \let\appsubsecentry = \numsecentry
- \let\unnsubsecentry = \numsecentry
- \let\numsubsubsecentry = \numsecentry
- \let\appsubsubsecentry = \numsecentry
- \let\unnsubsubsecentry = \numsecentry
+ % Wwe want section
+ % headers in our short contents
+ \def\nullentry##1##2##3##4{}
+ % \def\numsecentry##1##2##3##4{}
+ % \let\appsecentry = \numsecentry
+ % \let\unnsecentry = \numsecentry
+ \let\numsubsecentry = \nullentry
+ \let\appsubsecentry = \nullentry
+ \let\unnsubsecentry = \nullentry
+ \let\numsubsubsecentry = \nullentry
+ \let\appsubsubsecentry = \nullentry
+ \let\unnsubsubsecentry = \nullentry
\openin 1 \tocreadfilename\space
\ifeof 1 \else
\readtocfile
\setcounter{secnumdepth}{1}
(or\setcounter{secnumdepth}{2}
depending on your document class) should do the trick. See @egreg's answer for more explanations.