You won't lose multiple 1st or 2nd round picks. Compensatory picks are protected. This is where I think they failed when they changed the rules in the last CBA. Big market teams can still sign multiple FAs and take advantage of the system.
Let's use 3 type A players as an example. A team signs 3, they lose their natural 1st, 2nd, and 3rd round picks. They let 3 type A players walk, they gain 2 1st round picks and a 2nd (figure at least 1 team signing their player didn't have a 1st from another move or it was protected, ie top 15) plus 3 1s picks. Let's further assume their draft position is 20 and there were 30 supplemental picks awarded.
The first round is 30 picks - they have 2 top 30's.
The supplemental round is 30 picks - they have 5 top 60's.
The second round is 30 picks - they have 6 top 90's.
The 2s round is another 20 picks, the 3rd round is another 20 until they pick, so they gave up 3 top 130's. In exchange for 3 top 130's, they get 6 top 90's.
What they should have done is made all picks, natural picks and compesatory picks, subject to the rules of compensation. This would do a much better job of mitigating the loophole. They could still improve their position through the supplemental rounds, but not to the degree that they can now. The way it is now, depending on who becomes available, a big market team can improve both on the field and in the draft at the same time. It runs counter to the intent of the compensation system.
Supposing they did change the system to what I posted, let's look at what would have happened with the example.
The 3 players signed would have come at the expense of their natural 1st round pick and the 2 1st round picks they were awarded. They would still get the 3 1s picks. They would have 2 in the 2nd, their natural and the awarded one, and they'd keep their 3rd round pick. It's now 6 top 130 picks, with none in the top 30, vs. 6 in the top 90 with 2 in the top 30. Much better, if you ask me.