This is because, when the federal government offers money to the states, it can much more thoroughly, even if not directly, leech away the independence of the states. In order to offer these states funds the federal government must first tax the citizens of those states to collect the revenue that it plans on doling out to the states. The federal government then "offers" this money, that the states could have collected on their own if they wished to, back to the states on the condition that they spend it as the federal government wishes, and so long as they do whatever else the federal government wants them to do as well. For example, the federal government did exactly this with road construction and maintenance funds and speed limits. If states refused to post speed limits (which is understood in Constitutional law to be well within the sole jurisdiction of the states) in accordance to the "request" of the federal government they would not be given the funding that all the rest of the states were given. This would, in effect, would force that state to subsidize the road construction of the rest of the sates if they refused to give up their independence, and this would be very likely to greatly rile the voting populace of this state who don't appreciate having to bear that burden. This makes preserving state independence political suicide and pragmatically, impossible to go though with.
Further, if the states were left on their own, they very well might decide that they don't NEED so many roads, or to spend so much on any other particular program that the federal government funds for them. States, without federal government funding programs, would compete to give their constituents the most efficient amount of services for the cost. They wouldn't want to waste money building too many unnecessary roads or performing too much maintenance on them, but they also wouldn't want to spend too little, doing either would be likely to cost them precious votes. But when the federal government is doing out the cash the states have no incentive to limit their spending. They will always spend as much of those federal tax dollars as possible in order to maximize the amount of money flowing into their state and then they'll cry out for their state to get more afterwards.
For these reasons, it seems to me that it is funded federal programs that are pushed on the states, rather then the unfunded mandates, that pose the larger of the two problems to American society.