Your lawyer is correct that this is normal and it is an intentional feature of the LLC statute rather than a bug, that allows owners of LLCs to keep their ownership out of the public record, allowing a common owner, for example, to hide the fact that he is acquiring lots of property on the same block to assemble it into a larger unified parcel with more development possibilities.
Normally, the Operating Agreement adopted by the entity and the entity's books and records would establish you as the sole member of the LLC. The registered agent of the LLC, whose identity is a matter of public record, could establish that this was the correct initial Operating Agreement of the LLC and that you were affiliated with it, if necessary.
Another way to prove your ownership in the event of a dispute would be your tax records in which you paid income taxes on the rent earned every year on your personal IRS form 1040 and state income tax returns.
There is a tradeoff of certainty of proof of ownership for privacy involved in this situation, which is shared by many entities in U.S. law and is not particular to an LLC.
For example, a corporation does not list its shareholders as a matter of public record.
In contrast to the U.S., most countries in Europe, and most other countries with civil law system based upon continental European legal systems, require entity ownership in closely held firms to be formally filed in a notarized document held by a notary, which in those countries has responsibilities of a transactional lawyer, a the Secretary of State of a U.S. state, and a county clerk and recorder in a U.S. state, blended into one private sector position.
What would stop anyone else from writing their own affidavit claiming
to be the sole member of the LLC, then selling the houses, without my
It isn't impossible that someone could do that, but once you discovered that this was done, there would be a wealth of evidence from your signature on behalf of the LLC on leases, your connection to the address of the LLC, your bank statement, your tax returns, testimony from the registered agent and organizer, etc., that you were the true owner.
Then, the person who made the transfer would be prosecuted for fraud and go to prison, or at least, would be successfully sued and owe very substantial punitive and compensatory damages in an award that couldn't be discharged in bankruptcy.
Also, a signature on an affidavit long before a dispute arose, verified by a third-party notary, would also be more credible evidence than any recently generated documentation of the imposer owner of the LLC.
This is a reason why, in practice, LLC "identity theft" is very rare.
The far more common situation is that there are originally undisputed owners of the company and there is a dispute concerning transactions or events that purport to change ownership after the settled root situation.
For example, there might be a dispute over who paid for a membership interest if there is an erroneous booking entry in book keeping software, or a dispute over who the owner was when the funds that went into the entity came from a joint checking account of a husband and wife, or a dispute over who the effective owner of a company is when it is owned by a trustee and there is a dispute over who the correct successor trustee of a trust is in a particular case.