I don't think Congress would push for an override if Coolidge vetoes as he was initially inclined to do IOTL. Remember that the co-sponsors were both Republicans.
However, a bevy of restrictions remain in place due to all the xenophobic immigration acts from 1882 to 1917. Almost all of Asia (except the Western concessions on the Chinese east coast) is excluded from immigration by
the 1917 Act, as specified by
this map. The criteria for excluding immigrants is absurdly wide, with things like "professional beggars", "idiots" "feeble-minded", etc. Among those who supported it was the AFL, due to their worries about cheap foreign labor.
Here's the legislative history from 1875 to 1924.
1875 (Page): Barred Asian contract laborers, most Chinese women due to fears of (non-existent) Asian diseases.
Chinese Exclusion Act (1882): self-explanatory. Ironically (and stupidly, of course), it
voluntarily created a mass of illegal aliens, who did not receive citizenship until
1943. Chinese were not allowed to marry whites in California until
1948.
1882: head tax of 50 cents, or roughly $11 USD today.
1885: barred foreign contract workers.
1891: Commissioner of Immigration under Treasury, the first step towards creating the IRS.
Geary Act (1892): Renewal of the CEA, but a bevy of other restrictions that are too long to list.
1906: English proficiency requirement.
EQA of 1921: first numerical quotas.
You will not get all of these removed until McCarran-Walter in 1952 and mostly with the 1965 I&N Act.