Actually, resolution isn't a problem here. SARs that use narrow beam and high resolution are for detecting targets such as tanks or "small ships". Aircraft carrier is huge, which justifies the use of low resolution, allowing large area can be scanned quickly. Anything that is half the size isn't worth detecting anyway. Jamming signal will be detected by ELINT satellites, but ultimately needs to be countered by redundancies of multiple satellites. China will need to keep on launching Earth observation satellites.
resolution is a problem for any image sensor vs distance/area, there are more oil tankers, ships on the sea than a single carrier. in the radar image, you don't see the image of ship, you only see a spot. you are try to detect something in a million square mile area. when using narrow area SAR and high resolution, this mean revist, longer time. so basically divide a million square mile into thousands grid, and search each grid one by one. take long time, alot other ships in the area that are also big, also carrier is a moving target, which mean when you searching grid 2, the ship might already move to grid 1 from adjacent grid
Military radars are built with capabilities that civilian radars don't need. In the case of SARs, civilian version doesn't require the ability to detect/track moving targets, where as military version does. And as I have pointed out already, SARs are used on modern day battlefield for both mapping and detection/tracking of targets, showing your claim that SARs cannot detect/id ships as being false.
If there is one area that military sensors are better than civilian ones, it would definitely be in the field of radars. There is no such thing as civilian equivalents of APG-77 or SPY-1D radars.