You are misunderstanding the requirements. A will is not required to be notarized in New York State to be valid, and a will is not required to be filed prior to the death of the person writing it in New York State (and it usually isn't filed with anyone prior to death). Your steps 3 and 4 are wrong.
But, a will can not take effect after the death of someone who wrote it until the original Will (or a copy of the Will accompanied by testimony explaining why the original cannot be produced if the Will was not revoked, see Surrogate's Court Procedure Act § 1407) is filed with a court with probate jurisdiction (which in New York State is the Surrogate's Court) after the death of the person who wrote it.
The requirements are set forth at Estate, Powers and Trusts Law (EPTL) § 3-2.1. This states:
§ 3-2.1 Execution and attestation of wills; formal requirements
(a) Except for nuncupative and holographic wills authorized by 3-2.2,
every will must be in writing, and executed and attested in the
(1) It shall be signed at the end thereof by the testator or, in the
name of the testator, by another person in his presence and by his
direction, subject to the following:
(A) The presence of any matter following the testator's signature,
appearing on the will at the time of its execution, shall not
invalidate such matter preceding the signature as appeared on the will
at the time of its execution, except that such matter preceding the
signature shall not be given effect, in the discretion of the
surrogate, if it is so incomplete as not to be readily comprehensible
without the aid of matter which follows the signature, or if to give
effect to such matter preceding the signature would subvert the
testator's general plan for the disposition and administration of his
(B) No effect shall be given to any matter, other than the attestation
clause, which follows the signature of the testator, or to any matter
preceding such signature which was added subsequently to the execution
of the will.
(C) Any person who signs the testator's name to the will, as provided
in subparagraph (1), shall sign his own name and affix his residence
address to the will but shall not be counted as one of the necessary
attesting witnesses to the will. A will lacking the signature of the
person signing the testator's name shall not be given effect; pro-
vided, however, the failure of the person signing the testator's name
to affix his address shall not affect the validity of the will.
(2) The signature of the testator shall be affixed to the will in the
presence of each of the attesting witnesses, or shall be acknowledged
by the testator to each of them to have been affixed by him or by his
direction. The testator may either sign in the presence of, or
acknowledge his signature to each attesting witness separately.
(3) The testator shall, at some time during the ceremony or ceremonies
of execution and attestation, declare to each of the attesting
witnesses that the instrument to which his signature has been affixed
is his will.
(4) There shall be at least two attesting witnesses, who shall, within
one thirty day period, both attest the testator's signature, as
affixed or acknowledged in their presence, and at the request of the
testator, sign their names and affix their residence addresses at the
end of the will. There shall be a rebuttable presumption that the
thirty day requirement of the preceding sentence has been fulfilled.
The failure of a witness to affix his address shall not affect the
validity of the will.
(b) The procedure for the execution and attestation of wills need not
be followed in the precise order set forth in paragraph (a) so long as
all the requisite formalities are observed during a period of time in
which, satisfactorily to the surrogate, the ceremony or ceremonies of
execution and attestation continue.
It is better practice to notarize the Will when it is executed because this simplifies the task of proving that the person writing their wills and witnessing the wills did so at the time of the death of the person writing the will, which could be decades or more later. See Surrogate's Court Procedure Act § 1406. A will notarized at the time it is executed is sometimes called a "self-proving will". But notarization is not legally required for the will to be valid in New York State (or for that matter, in any other U.S. state).
A will is not generally filed with the Surrogate's court until after the death of the person who wrote it, and then it is usually filed in the Surrogate's court for the county in which the person who died was domiciled at death. (There are exceptions to the rule about which county the probate case should be filed in that are beyond the scope of this question and answer.)
There are two main exceptions to the rules cited above:
Members of the military, certain civilians attached to deployed military units, and mariners at sea, can make an oral will (called a "nuncupative will") to two witnesses, or can make a will entirely in his or her own handwriting (called a "holographic will") which is valid for one to three years after returning from that situation depending on the circumstances as set forth in EPTL Section 3-2.2. The requirements for a holographic will in New York State are more strict than in most other U.S. states.
Secondly, a will which was validly executed in the place where it was executed may often be recognized as valid at the time of death, (even if it wouldn't be validly executed under the law in force in New York State at the time of death), particularly if the person who died wasn't domiciled in New York State at the time when the will was signed, because that person had no good reason to think that New York State law would apply to the execution of the will. This is a "choice of law" issue and would be used to apply the law of another state or country where the will was signed and/or where the person who died was domiciled at the time the will was signed, rather than current New York State law.