First, the fact that they have the money to launch more satellites does not mean that they have the money to launch ALL the satellites that the full system needs. Indeed their projected cost for fully setting up Starlink is $10 billion dollars, which is more than the projected operating profit that they have had over their entire history. So it is easy to see that they won't be able to complete unless they get more money along the way.
In fact there is a significant risk that if they make less than they hope, that the company runs out of money and goes out of business. This subsidy reduces that risk, and therefore makes them more likely to complete, and likely to complete faster.
To achieve this purpose, the subsidy should be sized to help them over the hump, and not to pay all their costs. (Whether we should be handing out subsidies is a different question that I'm ignoring. Our government has chosen to, and the question is how to succeed.) Therefore there is nothing odd about the fact that it is less than 10% of the total amount of money that they think they will need. They aren't doing Starlink in the hopes of getting the subsidy. But the subsidy improves their odds of succeeding at Starlink.
And finally, you claim that the subsidy will only go directly to their bottom line. Which is to say that it shows up as profit and goes to Elon and other shareholders in the company.
It is true that plenty of companies operate that way. However after 20 years of running companies, it is extremely safe to say that Elon does not. In fact he tends to take every scrap of money that his company is generating, finds ways to borrow or raise more, and then reinvests it into even more ambitious projects.
So he won't be taking it as profit. What will he use it on?
Well, the three big projects that Elon is working on right now is building Starship, building Starlink, and preparing to settle Mars. Of those, Starship is required to finish Starlink at a price point where it is a viable business, and Starlink is required to fund Mars. Therefore it is essentially certain that he will pour this money directly into Starship and Starlink, rather than taking any of it as immediate profit.
Therefore we have good reason to believe that this money will be used for its intended purpose. We have good reason to believe that this money will in fact improve the odds of success, and enable them to deliver that success sooner. And therefore this money actually helps achieve the goal that the US government has set.