I would not assume that, especially since he has traveled back and forth to Venezuela. Considering his stature, most "bio" articles would note something along the line of "He was eventually naturalized in (fill in year)).
It takes a long time to become "naturalized" and until now, there really was no need to do so right away (particularly in his field with the established Visa system and "protection" coming from his industry).
The difference between Alvarado and Garcia is that Garcia didn't come to the U.S. as a jockey, he immigrated here to work and was drawn into horse handling and eventually being a jockey.
But Alvarado's father is a jockey in Venezuela and so the son followed in his footsteps to do the same, with Junior racing in both countries. And now you have 45 blowing up the whole Visa system (illegally).
From here - https://www.americasbestracing.net/jockeys/junior-alvarado
Alvarado was originally to be named Rafael Alvarado Jr. but when his father went to register the name it came out Junior Rafael Alvarado and he's been called Junior ever since. Alvarado and his wife, Kelly, have a son, Adrian, and a daughter, Adalyn.
Alvarado started riding in America in South Florida in 2007. He thought about returning to his native Venezuela but decided move his tack to Chicago at the urging of trainer Moises Yanez, for whom he enjoyed considerable success.
Junior made a big splash in his first meet at Arlington in 2008, winning 81 races to finish eighth in the standings. After the 2008 Arlington meet, he went to Hawthorne for a time before returning to Venezuela. He returned to the United States in time for the 2009 Arlington meet and jumped to the head of the class in his second meet at Arlington, taking the 2009 riding title with 110 wins, three more than James Graham. He won three stakes races at the meet in the Illinois Owners Stakes with Voy Por Uno Mas, the Sea Erin with Public Speaker and the Grade III Arlington-Washington Lassie aboard eventual Eclipse Award winning 2-year-old filly She Be Wild.
In his bid to repeat as riding champion, Junior finished second to Michael Baze in the standings with 97 victories. However, he scored the biggest win of his young career when he guided Richard L. Duchossois Éclair de Lune to victory in the Beverly D. Stakes to record his first Grade 1 win.
Alvarado moved in 2010 to New York to ride full time. Alvarado added Grade 1 wins in 2012 with Emma's Encore and 2013 aboard Strapping Groom and Flat Out. During the 2013 season, Alvarado estanblished (sic) a new benchmark for purse earnings in a single season with $12,865,602. In 2014, he won the $1.5 million Whitney Stakes on Moreno on Aug. 2 and later that month he recorded the 1,000th win of his career when he rode Sherkis to victory in the fourth race on Aug. 25 at Saratoga Race Course.
(snip)
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