Something like 40% of the Texian Army immigrated there from the US after the fighting started.... Sam Houston was a protege of Andrew Jackson. If you don't think the US was playing a "hidden hand" in Texas and was an entirely disinterested party, then you're kidding yourself.
Yes, but they were not sent by the Government itself.
No, these were volunteers, individuals who decided to make this journey on their own. Many even started out before the war broke out, because they felt they had nothing left in the US worth staying for.
Individuals like Davy Crockett. Who was defeated for reelection in 1835, and was a well known opponent of President Jackson. Who famously said upon his defeat "I told the people of my district that I would serve them as faithfully as I had done; but if not, they might go to hell, and I would go to Texas."
And along the way gathered up 30 more volunteers in Tennessee. Among them were the McCulloch brothers, neighbors of his in Tennessee. And they and the rest of the volunteers agreed to join the revolution, because they were promised 4,600 acres of land each (over 7 square miles). That is the reason why so many flocked to support Texas in the Revolution. All of the volunteers who went there from the US were offered extensive land grants. And those that served more than 3 months after 1935 were granted an additional 1,280 acres (total of 9.1 square miles). In all, over 1.3 million acres of land (2,000 square miles) was given to veterans of the Revolution.
No, there was no "US Government" involvement in the war. They largely had no interest in the conflict. But thousands of citizens flocked to the area because of the land bounties offered.
To make a comparison, consider one of the most well known Land Rushes in US history, that of Oklahoma. Where over 50,000 people traveled in the hopes of buying 160 acres each. And in a similar land rush in Oregon, settlers could make claims of up to 640 acres, which they bought at $1.25 an acre. But here was the new Republic of Texas, offering monumental land grants (almost 1,000 times larger) free of charge for simply going there to fight for them.
If people were willing to travel to Oklahoma or across the entire Continent to Oregon for 160-640 acres, it is not hard to imagine what people would have done in order to claim 4,600-5,880 acres.
Yes, people did indeed flood in from the US to fight. But for land. That is why between the time of the Revolution to 1850, the population soared from around 30,000 to over 200,000. Instead of returning home after the Revolution was won, those soldiers remained, as Texas citizens. If they were simply "US soldiers going to Texas" then they would have simply packed up and gone home after the war was won.
And finally, this was not even the first revolution in Texas. There was the earlier Fredonian Revolution in 1826 (just after the land grants to Anglos started, and 5 years after the Mexican Revolution). But this attempt ended in utter failure. The leaders offered no land for those who would fight for them, and not only did few come to their call for aid, most outright rejected them. In fact, Stephen Austin (the "Father of Texas") was approached to join and he soundly refused, stating that they were "deluding yourselves and this delusion will ruin you." And when Mexican authorities questioned Mr. Austin, he not only made an announcement that "infatuated madmen at Nacogdoches have declared independence."
He then raised a militia force of over 250 colonists to assist the Mexican Army in putting down the revolt.
I think you are missing some huge sections of both the history of the Texas Revolution, as well as the precursors that led to the revolution itself. Things that helped drive individuals like Stephen Austin to go from a loyal citizen of Mexico who actually raised soldiers to fight against an earlier Anglo revolution, to the leading founder that ultimately led the successful revolution a decade later.
It is the same kind of action that causes a highly decorated Colonel in the Colonial Militia to take up arms as the Commanding General of all American Forces in the US Revolutionary War. Because what happened in Mexico is not incredibly unlike what happened in the English Colonies that caused them to revolt. Something that was not present in 1826.