What is the safest way to escrow my money before purchasing bitcoins? While investigating the process of purchasing items with bitcoins, one site that I was learning about suggested, if you don't have bitcoins, visit LocalBitCoins.com and buy some.

OK, that made sense.  I don't have any bitcoins yet and have not yet invested in any mining hardware, so going someplace to buy some is necessary.

Then I started investigating the exchanges and how they work.

I'm accustomed to AmeriTrade.  I transfer money to AmeriTrade, they facilitate the buy-and-sale of stocks, etc.  Sellers know the money exists because Ameritrade has it and verifies it.  Buyers know they'll get what they're paying for because AmeriTrade has that, too.

However, that does not seem to be the case on LocalBitCoins.com.  It appears they don't want anything to do with escrowing the USD, only the BTC, and that appears to open them up to fraud.  It's one thing to hand my banking info to AmeriTrade, but to hand it to some one with a pseudonym who could be living anywhere in the world?

Question:&emsp;What is the most reliable way to escrow cash before making a BTC purchase?

It almost appears there isn't any.  People seem to want either face-to-face cash delivery, mailed cash delivery, Moneygram or Western Union transfers (ugh, expensive), or even gift card solutions.  All of this appears, well, mickey-mouse to the extreme.  It's like the old west where no one is willing to be "the bank."
Sites like localbitcoins.com (and eBay!) rely on reputation scores to assuage the type of concerns you express. 

If a user on localbitcoins.com has a high number of succesfully completed transactions, with a 100% overall score, I think you can feel reasonably reassured that this user is reliable and will do the right thing. They have a respectable track record of performing as expected.

If you are buying bitcoins, in fact you do not have to give your fiat (dollar) banking information out at all. Many bitcoin sellers will accept cash, either in person, or deposited at their bank - so they would have to give you their banking information.  Unfortunately, many traditional fiat banks are restricting cash deposits into user accounts from anyone but the named account holder.  (Perhaps because they sense their obsolescence versus the security, speed and de-centralized nature of cryptocurrency.) So, verify that the bank you are depositing into will accept cash deposits from third parties.
There are other ways to pay for the bitcoins you are buying, as you point out, such as "Zelle" (https://www.zellepay.com/how-it-works) and others. Check the localbitcoin.com adds for the different ways sellers of bitcoin may accept payment. 

Interesting that nobody wants to be "the bank". I think that is one of the beauties and reasons-for-being of crypto. There really is no need for banks, once you can receive, and send crypto on your own, from your own machine, with its own wallet. It obviates the need for third party intervention, really. Paying someone with crypto is like handing them cash, but it can be done electronically world-wide.

Maybe what you have here is a business opportunity for such an escrow company. One that will hold fiat deposits to secure transactions between seller &amp; buyers. It would receive fiat from buyers of crypto, and release the funds to the sellers of crypto when it verifies (by monitoring transactions in the blockchain) that the agreed-upon crypto has been deposited into the buyer's crypto address...

Crypto funds are "kept" by the coin's entire peer-to-peer network in public addresses, and exist everywhere worldwide where any peer in the coin's network keeps a copy of the blockchain: aka, the shared-ledger which accounts for the creation and transfer of coins. But only the owner of the each address should have the private key that can spend the funds in that address.
(Make sure to keep the private keys to your addresses truly private !)

It's even possible to print out your private key on paper, and erase any trace of it from a digital device, making it impossible for any hacker to abscond with your funds. Cryptocurrency is very similar to cash in this regard. Your private keys are literally the keys to your funds. If someone takes or copies your private keys, they can spend your crypto funds. This would be analogous to leaving fiat cash lying around somewhere where it can be stolen. 
Just keep those keys (in most wallet apps it's the 'wallet.dat' file in the .bitcoin directory) safe and secure and you will be fine.

Best of Luck