RootsChat.Com

a blogger's history of the place, but no information about lists of residents.
https://mixerrreviews.blogspot.com/2019/11/history-of-bexar-county-home-for-aged.html

these people might know more.
https://www.bexar.org/1322/Libraries-Archives  - Bexar County archives

https://www.mysapl.org/  (San Antonio Library)

I tried a few different ways on stevemorse.org also and didn't find him.
You said you found the list of ships that arrived from Hamburg in Oct 1907. Did you look at the Kaiserin Auguste Victoria on that list. That might be close enough for someone who wasn't Austrian.

The only other suggestion I have is to narrow down the time frame - when was he last seen in Galicia, when was he first seen in the U.S. - earlier than 1910? If he had children born in the US, when were they born?

newspapers.com for Annie Beck, 1880 to 1903, San Francisco.
reference to Annie (Senior)'s death in 1897, but you have that.

also, for schools:

https://www.newspapers.com/newspage/27532488/

San Francisco Chronicle
    Issue Date: Friday, June 17, 1887
    Page: 7

a listing for
Annie Beck, seventh grade
possibly at the Whittier Primary School
Miss N. A. Maloney, Teacher

Note find someone with a subscription. I am only trying to make sense out of an OCR scan and might be wrong.
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And this doc talks about SF schools during the fiscal year ending June 30, 1886,
listing for
Whittier Primary School - Harrison between 4th and 5th  < - would agree with Clara St location on the map

and says the Girl's HS location was on the north side of Bush Street between Hyde and Larkin

https://library.sfgenealogy.org/directory/san-francisco-county-directory/1887/1887_49.pdf

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Annie Junior wouldn't have entered high school until about age 13 or 14, unless things were different back then. I was 13.
7th grade at age 11 would be about right, but a little young.

edited to add: Or it might have been promotion /to/ 7th grade which would fit a little better. The article was about promotions, I think.

You can find an ED on the census if you know the address, at https://stevemorse.org/census/unified.html. First pick the year at the top of the page.

In 1900:
*** They are not at 52 Dominick Street in ED 11, their 1898 address
*** They are not at 311 Spring Street in ED 40, their 1902 address

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And I looked for anyone named Sullivan arriving in NYC in July 1873 and did not find them. I wonder if they are on the manifest with only their initials?

I think the letter S with the certificate means special, which I think is for marriages where the certificates were created months to years after the date of marriage.  I am not positive, but I think they are not available anywhere online.

I think S-415 is the Soundex code for Sullivan. the Italian Genealogy site includes them.