I disagree. J-20 is a VLO design with true 4S capability. But it's definitely designed with multirole in mind.
Being multirole means the aircraft, when necessary, can perform something else inadditional to, let's say, fighting for air-superority. But the aircraft is still dedicated to fight for air-superority. Being multirole doesn't mean the aircraft will be good at everything because that would clearly be an engineering impossibility.
This is by no means a bad thing. CMC needs something that can project power deep into South China Sea and even Malacca Straits.
I have pointed this out before, there are other aircraft that China can use to project power into South China Sea. Using J-20 here is an overkill.
In particular, the wings are high-mounted to give maximum space for an anti-ship missile.
High-mounted wings do not necessary mean anti-ship missile will be carried underneath.
A J-20 has a better chance at evading SPY-1 on the Arleigh Burke compared to a cruise missile.
You have no proof for this.
It has a better chance to avoid being tracked by SPG-62. It can get close to a Hawkeye before getting detected.
Which is another reason why the J-20 is an air-superority fighter. With an A2A load out, the J-20 can blind a CVBG's eyes in the sky, and make it easier for older generation of fighter-bomber to perform strike missions afterward.
To a CVBG, a regiment of incoming J-20 carrying short-range, high-speed ASM is more dangerous than facing the same number of long-range ASM fired at stand-off distances by H-6.
As somebody has pointed out already, small ASM's aren't going to do much good because they are too small. For your idea to work, there will need to be alot of J-20's in each sortie to carry a lot of missiles. China isn't going to have that much money to maintain that many J-20's, so your scenario doesn't come anywhere close to the reality on this count alone. In addition, by getting close to your opponent you are essently compromising your stealth advantage. You make no sense at all. If one can launch missiles from stand-off range, then there is no point in risking high-value assets to do it at close range.
I can also see J-20 loitering over a battlefield in a Taiwan scenario hitting mobile targets like Patriot batteries and radars with PGM. And if an F-22 arrives from Guam, well, it can hold its own too!
And when F-22 comes, those PGM will become liability in your scenario. Not only could they not be used on aircraft, but they will become weight penalty in an A2A engagement. Dealing with the F-22 is another reason why the J-20 isn't going to be a fighter-bomber.
I think something more blended wing with smaller vertical stabilizer would be even better at air superiority than J-20 but then it wouldn't be multirole, so for China's purposes J-20 is more pressing.
Yes, J-20 is pressing because maintaining air supermacy in the region is one of the most pressing matter for China. It also makes more sense to build an air-superority fighter first, because older generation fighter-bomber can be used for anti-surface missions once the sky has been cleared. Should a 4th generation fighter-bomber come online future, the J-20 can also be used to provide escort.