I was recently looking for a simple, robust, usable, non-branded swf video player to use as a flash fallback when using the HTML5 <video> element, and I found the Strobe Media Playback player from Adobe. It fills all those requirements, and is fairly feature-rich. The files are directly downloadable from their Sourceforge directory. Here, more or less, is how I use it:
<video controls poster="video.jpg">
    <source src="video.mp4" type="video/mp4">
    <source src="video.webm" type="video/webm">
    <object type="application/x-shockwave-flash" data="StrobeMediaPlayback.swf">
        <param name="movie" value="StrobeMediaPlayback.swf">
        <param name="flashvars" value="src=video.mp4&poster=video.jpg">
        <img src="video.jpg" alt="Video">
        <p>Your browser can’t play HTML5 or flash video. <a href="video.mp4">Download it</a> instead.</p>
    </object>
</video>
It supports lots of parameters specifiable from within the flashvars parameter, such as autoplay and a whole host of parameters related to the controls (the documentation lists them).