In brief, yeah sure we could be at our NAIRU now (i severely doubt it, but anyway...), and thus no contradiction in your arguement if participation rises now. But unemployment has been falling for years and labour participation rising as well. We can hardly have been at the NAIRU for all that time. Even though it can change, it's not going to come down that quick. Thus the contradiction remains-really hasn't changed at all. And for example, U was 3.9% a few months ago, about 4.3% now i think. Does that mean our NAIRU was 3.9% at the time? And has it jumped up to 4.3% now? And what increase in structural problems caused that?
Apart from that, the scenario you described:
AD is stimulated at the NAIRU
Business demand more labour. No efficient labour available so they hire the next best thing- inefficient labour (the structurally unemployed who lack sufficient skills to be employed under usual circumstances- but just about everyone has some basic skills in everything-but as they lack 'sufficient skills' to do the job properly, there are still structural problems)
Inefficient labour causes average cost of production to rise, inflation increases, businesses shed back to the NAIRU
Is precisely the long run phillips curve- smart man Friedman-coherently explain why it's wrong