Try this updated code in your system..
It will produce a table centered vertically in the frame.
If this is not the case, check if any packages need to be updated.
Added \listfiles
to list in the .log
file all the files loaded and compare with my output.
The table
environment adds unnecessary spaces.
Also corrected measure
to compile you code. measure = vbox
is needed for varwidth
to measure the width of the cells when they contains vertical material, as in this case.
Note that after =\vspace{-\baselineskip}
fails to work properly and you get warnings from varwidth
, but after=\vspace{<dimension>}
works fine. This is the reason to use \vshift
: \the\vshift
will result in -13.6pt
, the (-) baseline skip.

I changed the columns specs but this is not related with the question.
% !TeX TS-program = pdflatex
\listfiles
\documentclass {beamer}
\mode<beamer>{\usetheme{Madrid}}
\title[Test Title]{Test}
%\usepackage{systeme}
\usepackage{tabularray}
\UseTblrLibrary{booktabs, varwidth}
\usepackage{enumitem}
\newlength{\vshift}% added
\setlength{\vshift}{-\baselineskip}
\setlist[itemize]{
nosep,
leftmargin=*,
label=\tiny\color{black}$\blacksquare$,
after = \vspace{\the\vshift},% <--- changed
before=\vspace{\the\vshift}, % <---
}
\begin{document}
\begin{frame}{Title}
\framesubtitle{Subtitle}
\begin{table}[!htb]
\small
\begin{tblr}{hlines, vlines,
colspec = {Q[1,c, font=\bfseries] X[2,c,m] X[2,c,m]}, % changed
row{1} = {font=\bfseries},
measure = vbox } % changed <<<<<<<<<<<<<<<<
One & Two & Three \\
Method 1
& \begin{itemize}
\item text
\item text
\item text
\item text
\end{itemize}
& \begin{itemize}
\item text
\item text
\item text
\item text
\end{itemize} \\
Method 2
& \begin{itemize}
\item text
\item text
\item text
\item text
\end{itemize}
& \begin{itemize}
\item text
\item text
\item text
\item text
\end{itemize} \\
Method 3
& \begin{itemize}
\item text
\item text
\item text
\item text
\end{itemize}
& \begin{itemize}
\item text
\item text
\item text
\item text
\end{itemize} \\
Method 4
& \begin{itemize}
\item text
\item text
\item text
\end{itemize}
& \begin{itemize}
\item text
\item text
\item text
\end{itemize} \\
\end{tblr}
\end{table}
\end{frame}
\end{document}
Output of the .log file
First lines
This is pdfTeX, Version 3.141592653-2.6-1.40.22 (MiKTeX 21.3) (preloaded format=pdflatex 2021.10.12) 30 NOV 2021 00:25
entering extended mode
**./myfile.tex myfile.tex LaTeX2e <2020-10-01> patch level 4
L3 programming layer <2021-02-18>
At the end of the file
*File List*
beamer.cls 2021/03/19 v3.62 A class for typesetting presentations
beamerbasemodes.sty
etoolbox.sty 2020/10/05 v2.5k e-TeX tools for LaTeX (JAW)
beamerbasedecode.sty
ifpdf.sty 2019/10/25 v3.4 ifpdf legacy package. Use iftex instead.
iftex.sty 2020/03/06 v1.0d TeX engine tests
beamerbaseoptions.sty
keyval.sty 2014/10/28 v1.15 key=value parser (DPC)
geometry.sty 2020/01/02 v5.9 Page Geometry
ifvtex.sty 2019/10/25 v1.7 ifvtex legacy package. Use iftex instead.
geometry.cfg
size11.clo 2020/04/10 v1.4m Standard LaTeX file (size option)
pgfcore.sty 2021/05/15 v3.1.9a (3.1.9a)
graphicx.sty 2020/09/09 v1.2b Enhanced LaTeX Graphics (DPC,SPQR)
graphics.sty 2020/08/30 v1.4c Standard LaTeX Graphics (DPC,SPQR)
trig.sty 2016/01/03 v1.10 sin cos tan (DPC)
graphics.cfg 2016/06/04 v1.11 sample graphics configuration
pdftex.def 2020/10/05 v1.2a Graphics/color driver for pdftex
pgfsys.sty 2021/05/15 v3.1.9a (3.1.9a)
pgfrcs.sty 2021/05/15 v3.1.9a (3.1.9a)
pgfrcs.code.tex
pgfsys.code.tex
pgfsyssoftpath.code.tex 2021/05/15 v3.1.9a (3.1.9a)
pgfsysprotocol.code.tex 2021/05/15 v3.1.9a (3.1.9a)
xcolor.sty 2016/05/11 v2.12 LaTeX color extensions (UK)
color.cfg 2016/01/02 v1.6 sample color configuration
pgfcore.code.tex
xxcolor.sty 2003/10/24 ver 0.1
atbegshi-ltx.sty 2020/08/17 v1.0a Emulation of the original atbegshi package
with kernel methods
hyperref.sty 2021-02-27 v7.00k Hypertext links for LaTeX
ltxcmds.sty 2020-05-10 v1.25 LaTeX kernel commands for general use (HO)
pdftexcmds.sty 2020-06-27 v0.33 Utility functions of pdfTeX for LuaTeX (HO)
infwarerr.sty 2019/12/03 v1.5 Providing info/warning/error messages (HO)
kvsetkeys.sty 2019/12/15 v1.18 Key value parser (HO)
kvdefinekeys.sty 2019-12-19 v1.6 Define keys (HO)
pdfescape.sty 2019/12/09 v1.15 Implements pdfTeX's escape features (HO)
hycolor.sty 2020-01-27 v1.10 Color options for hyperref/bookmark (HO)
letltxmacro.sty 2019/12/03 v1.6 Let assignment for LaTeX macros (HO)
auxhook.sty 2019-12-17 v1.6 Hooks for auxiliary files (HO)
kvoptions.sty 2020-10-07 v3.14 Key value format for package options (HO)
pd1enc.def 2021-02-27 v7.00k Hyperref: PDFDocEncoding definition (HO)
hyperref-langpatches.def 2021-02-27 v7.00k Hyperref: patches for babel languages
intcalc.sty 2019/12/15 v1.3 Expandable calculations with integers (HO)
etexcmds.sty 2019/12/15 v1.7 Avoid name clashes with e-TeX commands (HO)
puenc.def 2021-02-27 v7.00k Hyperref: PDF Unicode definition (HO)
url.sty 2013/09/16 ver 3.4 Verb mode for urls, etc.
bitset.sty 2019/12/09 v1.3 Handle bit-vector datatype (HO)
bigintcalc.sty 2019/12/15 v1.5 Expandable calculations on big integers (HO)
hpdftex.def 2021-02-27 v7.00k Hyperref driver for pdfTeX
atveryend-ltx.sty 2020/08/19 v1.0a Emulation of the original atvery package
with kernel methods
rerunfilecheck.sty 2019/12/05 v1.9 Rerun checks for auxiliary files (HO)
uniquecounter.sty 2019/12/15 v1.4 Provide unlimited unique counter (HO)
beamerbaserequires.sty
beamerbasecompatibility.sty
beamerbasefont.sty
amssymb.sty 2013/01/14 v3.01 AMS font symbols
amsfonts.sty 2013/01/14 v3.01 Basic AMSFonts support
sansmathaccent.sty 2020/01/31
scrlfile.sty 2021/03/17 v3.33 KOMA-Script package (file load hooks)
scrlfile-hook.sty 2021/03/17 v3.33 KOMA-Script package (using LaTeX hooks)
scrlogo.sty 2021/03/17 v3.33 KOMA-Script package (logo)
beamerbasetranslator.sty
translator.sty 2020-08-03 v1.12c Easy translation of strings in LaTeX
beamerbasemisc.sty
beamerbasetwoscreens.sty
beamerbaseoverlay.sty
beamerbasetitle.sty
beamerbasesection.sty
beamerbaseframe.sty
beamerbaseverbatim.sty
beamerbaseframesize.sty
beamerbaseframecomponents.sty
beamerbasecolor.sty
beamerbasenotes.sty
beamerbasetoc.sty
beamerbasetemplates.sty
beamerbaseauxtemplates.sty
beamerbaseboxes.sty
beamerbaselocalstructure.sty
enumerate.sty 2015/07/23 v3.00 enumerate extensions (DPC)
beamerbasenavigation.sty
beamerbasetheorems.sty
amsmath.sty 2020/09/23 v2.17i AMS math features
amstext.sty 2000/06/29 v2.01 AMS text
amsgen.sty 1999/11/30 v2.0 generic functions
amsbsy.sty 1999/11/29 v1.2d Bold Symbols
amsopn.sty 2016/03/08 v2.02 operator names
amsthm.sty 2020/05/29 v2.20.6
beamerbasethemes.sty
beamerthemedefault.sty
beamerfontthemedefault.sty
beamercolorthemedefault.sty
beamerinnerthemedefault.sty
beamerouterthemedefault.sty
beamerthemeMadrid.sty
beamercolorthemewhale.sty
beamercolorthemeorchid.sty
beamerinnerthemerounded.sty
beamerouterthemeinfolines.sty
tabularray.sty 2021-09-01 v2021N Typeset tabulars and arrays with LaTeX3
expl3.sty 2021-02-18 L3 programming layer (loader)
l3backend-pdftex.def 2021-03-18 L3 backend support: PDF output (pdfTeX)
xparse.sty 2021-01-09 L3 Experimental document command parser
xparse-2020-10-01.sty
booktabs.sty 2020/01/12 v1.61803398 Publication quality tables
varwidth.sty 2009/03/30 ver 0.92; Variable-width minipages
enumitem.sty 2019/06/20 v3.9 Customized lists
supp-pdf.mkii
epstopdf-base.sty 2020-01-24 v2.11 Base part for package epstopdf
epstopdf-sys.cfg 2021/03/18 v2.0 Configuration of epstopdf for MiKTeX
nameref.sty 2021-04-02 v2.47 Cross-referencing by name of section
refcount.sty 2019/12/15 v3.6 Data extraction from label references (HO)
gettitlestring.sty 2019/12/15 v1.6 Cleanup title references (HO)
q956v3.out
q956v3.out
translator-basic-dictionary-English.dict
translator-bibliography-dictionary-English.dict
translator-environment-dictionary-English.dict
translator-months-dictionary-English.dict
translator-numbers-dictionary-English.dict
translator-theorem-dictionary-English.dict
ninecolors.sty 2021/05/24 v2021C Select colors with proper color contrast
umsa.fd 2013/01/14 v3.01 AMS symbols A
umsb.fd 2013/01/14 v3.01 AMS symbols B
ot1mathkerncmss.fd 2020/01/31 Fontinst v1.933 font definitions for OT1/mathkerncmss.
***********
\begin{table}[!htb]
...\vspace{-1cm} \begin{table} \small...