I think any solution that doesn't involve easily obtainable temporary worker status is doomed to fail. Part of my solution is to create a temporary worker visa that any employer can apply for easily. With this temporary legal status you can get SSN, driver's license, and all the goodies but not unemployment insurance. Your status would depend on your employment just like H visa holders, so if you're not working and just causing trouble, you're not legal. You'll be given 90 days to change jobs, after which you'll be out of status. After you accumulate a certain number of years (say 10) of being a good member of society (no felony conviction) and a good taxpayer, you would be eligible to adjust status to a permanent resident (green card).

The key here is to make the visa easy and inexpensive to get by employers. If a mom & pop store can get you legal status, there really is no excuse to being illegal. This way the government is justified in setting up strong enforcement.

The only amnesty involved would be to allow current undocumented immigrants to sign up for the program by paying a modest fine, say $2000. FYI, current official fee (not fine) for adjustment of status to get a green card is $1070.