Modify footer used by elsarticle.cls

You have put your definition before the call to elsarticle, so the newer one overrode the older one. You need to change the order of definitions!

This works for me:

\documentclass{elsarticle}
\usepackage{lipsum}
\makeatletter
\def\ps@pprintTitle{%
 \let\@oddhead\@empty
 \let\@evenhead\@empty
 \def\@oddfoot{}%
 \let\@evenfoot\@oddfoot}
\makeatother
\begin{document}     
\title{Paper}
\author{A. U. Thor}
\date{\today}
\maketitle
\lipsum[1-2]
\end{document}

Or you can replace the preprint footnotes with page number, so as to make the first page footnote style consistent with the following pages. To achieve this, simply:

\documentclass[review]{elsarticle}
\makeatletter
\def\ps@pprintTitle{%
 \let\@oddhead\@empty
 \let\@evenhead\@empty
 \def\@oddfoot{\centerline{\thepage}}%
 \let\@evenfoot\@oddfoot}
\makeatother

You could try the following code, which uses the instruction \patchcmd provided by the etoolbox package. (What the code does is to replace the long argument of the definition of @oddfoot with \relax...)

\documentclass{elsarticle}
\usepackage{etoolbox}
\makeatletter
\patchcmd{\ps@pprintTitle}{\footnotesize\itshape
       Preprint submitted to \ifx\@journal\@empty Elsevier
       \else\@journal\fi\hfill\today}{\relax}{}{}
\makeatother
[rest of preamble]