Even assuming (probably incorrectly) that the (very cheap, low powered) Fire is capable of running ICS well, most people don't want to flash ROMs even if it is easy. They worry about it going wrong, they worry about warranties and support.
And most people neither know nor care what ICS is. They want to read some books, surf the web and watch some movies. They're not going to see why ICS is better.
And people like the close Amazon content integration which this would remove. They like dealing with a retailer they can trust in ways that are pain free. Google have a very limited history as a retailer. Even if they put their own content delivery in place of Amazon's people don't know and trust them in this space in the same way they trust Amazon.
This move would give a small number of geeks access to a low cost ICS tablet, it would do nothing to turn around the vast majority of Fire users.
There is already an ICS ROM for the Kindle Fire. It's pre-release -has touch, wifi, sound but no gpu acceleration yet- so it's probably not a good indicator of performance. You can find a build as well videos on xda: http://forum.xda-developers.com/forumdisplay.php?f=1309
Out of curiosity, does 'competition' always mean 'enemy'?
"... it can focus on its main enemy – the Apple iPad"
Google knew, or should have known, when they released Android the way they did that something like this might happen.
There is already an easy process to install ROMs on the Kindle Fire, including a per-alpha ICS one. The tool is here: http://teamw.in/project/twrp
A guide to use it is here: http://techwalls.com/gadget/how-to-install-android-4-0-ice-c... (workaounds no longer needed)
The ROM is being developed on XDA: http://forum.xda-developers.com/showthread.php?t=1379208&...
Video of pre-alpha ROM: http://moooom.dyndns.info/?p=107