Removal strategies vary depending on what equipment you already have, can negotiate to borrow, or are willing to rent or buy.
The pipe stubs don't leave much to grip but if you or a neighbor (etc) can get a welder there, a chain could be attached to each pipe. That provides an attachment for lifting. Otherwise, it does work to tie chain around an object just the same as one could do with rope. A long towing chain could be tied around the concrete in a variety of ways. The block could then be lifted with compact heavy equipment like a skid loader or mini excavator.
In case heavy equipment goes a little too far there are good alternatives. The slab will have to be broken first though so the posts can be dealt with one at a time. Consider upgrading the attack to a 60 pound electric demolition hammer from a tool rental store. That's generally the last option short of renting a towable air compressor and pneumatic jack hammer. The powered hammers aren't likely to crack the concrete clear through, but either will chip away at it faster than the sledge does.
With the slab broken each post might be lifted out with simpler tools. A chain hoist hung from a frame, such as 2x4 lumber lashed into a tripod, could work (I've pulled up many fence posts with a 1 ton Harbor Freight chain hoist and a lashed tripod). An engine hoist might do the job as well.
In case the posts must be dug out manually, here's one last suggestion: use a large wet/dry vacuum. Loosen the dirt with a digging chisel and vacuum the loose stuff out of the hole. You'll get away with removing less soil as compared to digging with a spade and, for the depth you have to go, it could be less work too.