This solution might not be the best since it hides the video title along with the youtube controls (may be undesirable) but some people might find it useful.
You can make the video player really tall and then crop the top and bottom to hide the more videos section, which is done with divs, so our embed html may look something like this:
<body>
<div id="player-size" style="">
<div id="cropping-div" style="">
<div id="div-to-crop" style="">
<div id="player-wrapper">
<!-- 1. The <iframe> (and video
player) will replace this <div>
tag. -->
<div id="player"></div>
</div>
</div>
</div>
</div>
<script async src="/youtube-player.js"></script>
</body>
</html>
demo: here
original answer: here
Perhaps the code may be modified to only hide the bottom portion of the player only but that will still hide the player controls as well. It is worth mentioning that custom player controls can be made using the player.seek() methods from the iFrame API YouTube has.