The simple answer is Yes. The best scripture for this comes from Hosea, because this is the only example of a prophet claiming the nation ( or part of it ) is not "my people" (Hosea 1:10). However, Hosea says in the next chapter "The Lord will say You are my people". ( Hosea 2:23 ). The only thing that Hosea says is needed is "return". No new heart; no new spirit, no salvation. Just return. ( Hosea 3:5, Hosea 6:1, Hosea 12:6, Hosea 14:1, hosea 14:2, Hosea 14:7 ). So that's according to Hosea.
Zooming out, do a search for "my people" on Tanach. What you'll find is that the Jewish people are called "my people" throughout the canon, regardless of their level of observence. Jeremiah, for example speaks of the LORD rebuking "my people", and there can hardly be a harsher, more unholy period in the Jewish narrative. Yet still they are God's people. ( Jeremiah 18:15 ).
Also, a minor mention: Isaiah 44:7 - at some point God appointed an "eternal people". Although the KJV translates it to "ancient people".
Now looking specifically at what Ezekiel is saying. Start at verse 13.
13: Ezekiel asks "Lord God, are You making a complete end to the remnant of Israel?"
14: God answers
15: "Your brothers and kinsmen, the remnant, were told to leave"
16: "I, God, removed them and scattered them"
17: Says God "I will gather and give them the land"
Note: no new spirit is needed for the remnant to return to the land
18: "The remnant will come to the land and purify it"
19: "I, God, will give a new heart and new spirit, and they will be my people"
20: "So that they will keep my statutes etc.."
So, according to Ezekiel 11, God remains attached to a remnant, and this remnant while scattered is not a "people". Part of going from remnant to people involves getting a new heart and spirit. But also living in the land. I am confident that there are Jewish people in Israel who live up to these standards. So maybe they are considered God's people, and I'm not. But I can still be in God's good graces with out the new spirit, new heart.
So, no, a new spirit really isn't needed in spite of Ezekiel 11. Looking at the whole canon, and especially Hosea shows that. Ezekiel appears to be an outlier.
Well, Ezekiel takes this prophecy and broadens the horizon. In Ezekiel 18, and again in 36 and 37, we get a repetition of the call.
Ezekiel 37:25, 'Then will l sprinkle clean water upon you, and ye shall be clean: from all your filthiness, and from all your idols, will l cleanse you.
A new heart also will l give you, and a new spirit will l put within you: and l will take away the stony heart out of your flesh, and l will give you an heart of flesh.'
In chapter 37 we have the resurrection of the dry bones.
37:13,14. 'And ye shall know that l
am the LORD, when l have opened your graves, O my people, and brought you up out of your graves,
And shall put my spirit in you, and ye shall live, and l shall place you in your own land: then shall ye know that l the LORD have spoken
it, and performed
it, saith the LORD'.
So, who is 'the resurrection, and the life'? See Matthew 27:52.
The unbroken nature of the scriptures allows us to see a repetition of the key teachings. In Jeremiah 31:31-34, the Lord repeats his teaching:
v.33.' But this shall be the covenant that l will make with the house of lsrael; After those days, saith the LORD, l will put my law in their inward parts, and write it in their hearts; and will be their God, and they shall be my people'.
Now, the story of David taking the ark to Jerusalem, 2 Samuel 6, may strike you as totally unrelated to the theme of the new covenant, but please look again.
David sets the ark (containing the Word) upon a 'new cart'.
David sang a new song, and danced, as they bore the ark to the city of David.
'And as the ark of the LORD came into the city of David, Michal Saul's daughter looked through a window, and saw king David leaping and dancing before the LORD; and she despised him in her heart'.
Who do you think Saul, and his daughter, Michal, represent?