The relay networlk already doesnt do full verification. It does simple SPV validation simply as a DoS precaution, but thats already rather quick. Building blocks on top of another miner without validation is still not possible no matter what precautions you take...Imagine a pool op of a smaller pool who wants to have some fun...they break into a large, "trusted" miner and change it to include some invalid transaction in its blocks, now it finds a block and everyone else goes off and mines on it, giving the smaller miner a chance to mine blocks for free while the rest of the network is off building invalid crap.
If a big pool got hacked you might be mining on invalid data from the time you submit their block to your local bitcoind and until you get the response back that it rejects the block. After that you switch to building on the previous parent block and mark this pool as untrusted so it won't be repeated.
About 100 times per day you get a sub-second advantage to avoid creating an orphan race. And very rarely, probably less often than once a year or maybe never, you mine invalid data for a sub-second. For every invalid block you mine, you have avoided 36 500 orphan races, of which you would have lost more than half because the other guy had a head start. Sacrifice one block (or likely zero) to save 20 000.
SPV clients are counting the blocks you produce to be valid. Setting up that kind of verification also produces extreme fragility in that if a software error makes a good node produce a bad block you could have a whole series of bad blocks created by mutually trusting hosts, creating a large fork in the network. This kind of approach is also vulnerable to attack since without a PKI infrastructure you cannot determine who the source of a block is— for example it could just be a sybil who is relaying good blocks to you from other parties but later starts feeding you trash. (and, of course, imposing identities on mining has a multitude of problems and risks that go beyond the simply technical)
In the rare event that an invalid block might be created, it's not going to reach SPV clients.
Verification doesn't need to be 100%. You can check that the difficulty (bits) is correct, that the generation transaction pays to a known address used by a "trusted" pool, and that the block hash is valid at this difficulty. So an attacker could mine an invalid block and cause you to mine invalid data for less than a second. That's a cost of over 25 BTC to them. They could repeat it for every trusted payment address you have, but no more than that.
Say you trust 4 sources. An attacker could spend over 100 BTC to make you mine invalid data for maybe one second in total. A smarter attacker would use blockwithholding instead, making you lose not 1 second of mining but 4 blocks (average) worth of mining effort, which is much much more than any pool does in 1 second.
Please don't do this. It undermines the security model of Bitcoin, and at most saves you a few _milliseconds_ (and even there it isn't lost work because you could theoretically find a block and win the block race) since virtually all the transactions are already verified, and cached in memory and the node doesn't check them again.
I don't see how it undermines the security model of Bitcoin. If you mine a block before the parent is verified by your own bitcoind you could wait until that happens before sending your new block to the relay network. That way an invalid block never leaves your pool. This way others on the relay network will only see the first invalid block from someone else. You will never add another invalid that they will see.
Say you get blocks from trusted sources 100 times per day and it saves you a total of 25 seconds of mining where you would have entered an orphan race and the other guy has a head start. Maybe 15 seconds where you would have lost an orphan race if you had gotten a block. That's an hour and a half of worthless mining saved per year.
You are right there isn't much to save. For most pools 1.5 hours of mining per year isn't a lot of blocks. While I'm pretty sure this would be of help and not do any harm, the benefit probably doesn't outweigh the effort of implementing this, which is why it has just stayed an idea.