Agreed, teaching illegally in China is not to be recommended, but some foreigners are doing it. Just to reiterate that teaching with an L or an F is illegal. Foreigners who arrive on L's, start teaching, and are awaiting going on a HK visa run are just as guilty as those who abuse the regulations for years on end. That should be stressed.
Arriving on an L, and looking for work, is dodgy. But maybe said foreigners were unable to obtain job offers from Chinese employers whilst in their homelands, making it impossible for them to arrive in China on a Z.
And other foreigners are getting Z visas because unscrupulous recruiters are forging other foreigners' university degrees to get Z visas for their unqualified jobseekers with or without their knowledge, and with or without the knowledge of the FAO's at certain educational establishments, both public and private! A recent post on the board highlighted a case where a recruiter offered a forged BA degree to an unqualified foreigner for 4800 Yuan! Said agent also used the electronic copy of the forgery to obtain a job for said foreigner, who was aware of the fraud, as was the school's FAO!

I forgot to mention another group, of which some of its members might be tempted to use agents for job hunting. That's the golden- oldies who are over 55, especially those over 60. Some of those between 55 and 59 may have had difficulty obtaining job offers due to a combination of ageism and mediocre resumes. Those over 60 because of enforcement, though not necessarily uniform, of an employment regulation on the books that can prevent the issue of Z visas to this group. Recent evidence suggests that said regulation is being strictly enforced in certain provinces.

And, finally, some foreigners are using forged Master's degree certificates and PhD's to enhance their salaries here. Maybe some agents are peddling said forgeries too! I wouldn't rule that out in China!