There are ways. Most illegals work for a business of some sort; there's only so many housekeeping jobs available. If you are a business owner and pay someone in cash off the books, you usually have to do some creative accounting. Basically, you have to engage in tax fraud. The IRS could detect most of this by doing more audits of businesses in idustries that tend to hire illegals. Most businesses only hire illegals if they have fraudulant papers so they don't have to engage in tax fraud. The INS can weed out close to 100% of these cases by more careful screening and verification of the documents.
They can be deported if and when they try to access government services or try to obtain free medical care.
There's more to it that that. It is already necessary to show a state or other government-issued ID, or have a social security number, in order to open bank accounts and engage in many every-day life activities. Prohibit all governments from granting of such IDs to illegals (currently illegals can get drivers' liscences in some states). Make it impossible to get a social security number if you are illegal (currently, it is not too difficult). Prohibit the use of the new matricula card. Stiffen the penalties for document fraud. Such measures would make the life of an illegal very difficult, and would likely encourage most of those who don't get caught to leave voluntarily.
I agree with all you said but I don't want to have to carry my government-issued ID at all times and be stopped for any reason to prove I can produce it.