This will be a pretty complicated discussion. By way of background I did title insurance claims throughout the US for various underwriters for about thirty years.
To begin with, as I understand it, notary publics in the US serve a different function than notaries in other countries. Their main function here is to take the acknowledgement of the signor of a document that the signature is theirs and to verify their identity through state approved identification. I believe notaries may be prohibited from advertising their services in Spanish to avoid confusion because of the different role they play in other countries, but I may be mistaken.
Each state has it's own requirements for the acceptable form of ID and record keeping requirements. Some states such as CA require the notary keep a log, some require a thumb print, some don't require any documentation. I've spoken with many notaries about questionable documents and have often been told while they don't remember that specific transaction they would never take an acknowledgment without proper ID. I've had others who have admitted their boss brought in a document signed by someone else but because the boss vouched for the authenticity of the signature went ahead and notarized it anyway. I've had title company employees tell me Mary is the only notary in the office so all the closers take the signed docs to her to notarize even though she never met the signor. It's far from a perfect system and is open to fraud and abuse and I agree with an earlier poster who has concerns about remote notarization. It will increase fraud in my opinion.
As far as searching a land registry I believe your describing a Torrens system where the state assures a real property purchaser of the title status of a particular property. One does not even search the title but applies for a title certificate which shows the property now being registered in the buyers name. If there's an error resulting is a loss the state pays the damaged party in accordance with state law. I believe there are only a handful of states where a Torrens system is used. The other states rely on an abstract system where counties, a subsidiary government body, is charged with accepting documents for recording and maintaining an index of them. The county does not examine the document for legal sufficiency of effect, only that they are entitled by statute to be recorded. The effect of recording is to provide constructive notice to third parties of the documents existence. As an aside, I believe most people are unaware that in in Florida, a document is considered recorded and provides constructive notice when it is deposited with the Clerk of the Court and is assigned a document number. It does not need to appear in the indices for it to impart notice and purchasers take subject to it.
One does not need to be an attorney to search, and more importantly in my opinion, examine title, but to do it correctly one needs time and training in the county and state where the property lies. A title examiner NY cannot start searching and examining title in Texas without significant training because each state and even each county has their own idiosyncrasies in recording and indexing which, if you don't know, can cause a defect in title to go unidentified.
Title insurance is not required for the purchase of property though to my knowledge most lenders will not loan money without being insured under a title policy. In my opinion it is foolish for anyone, even someone schooled in searching and examining title, to put significant money at risk without getting a policy because not only does it indemnify you for a loss related to a covered defect identifiable from the record, it also indemnifies for a covered loss which can't be found in the record such as fraud, forgery and incompetency. While many feel title insurance is not worth the money because they pay out relatively little in losses as compared to the premium, I believe that's because they spend much of their income in the search and examination process.
The closing of a purchase is frequently handled by a title company as a matter of convivence for the parties though it can be closed by a separate attorney with the title insurance issued separately. I believe that's required in NC. For some clarification there are generally two parties involved in the issuance of a title policy. First there is the policy issuing agent, the consumer facing part of the two. Then there is the title underwriter, who is financially responsible for receiving, investigating and paying covered claims. For further clarification, some agents are owned by an underwriter, they're referred to as direct shops. Then there are independently owned agents who are authorized to issue policies for one or more underwriters.
Escrow is a separate function that is frequently handled by the same title company, again as a matter of convenience though it can be handled by a separate escrow company. I believe that's common in CA. The escrow agent is responsible for collecting the purchase funds from the buyer and the buyer's lender if there is one and obtaining payoff's from lien holders and seeing to their payment, paying property taxes that may be owed and any other charges required to be paid. Either the title company or the escrow company should be responsible for the recording of the proper docs.
This is a brief outline of the process. There are hundreds of possible variations and thousands of things that can and do go wrong. I hope it helps and good luck.