There were three main things that people would have thought that the US would lose if some other country annexed the Philippines after the Spanish-American War.
First, people thought that the US would lose an economic opportunity. The Philippines had raw materials such as sugar, hemp, and coconut. These were things that could be valuable for the US economy. In addition, if the US took the Philippines, American companies could sell manufactured goods there. Both of these opportunities could have been lost if some other country had annexed the Philippines and closed off or dominated its trade.
Second, people thought that the US would lose an opportunity to project military power in Asia. At this time, a country needed to have naval bases around the world in order to really be powerful. The US had nothing west of Hawai’i (except Guam, which is much smaller and which it took at the same time that it took the Philippines) and so would not have been able to project power into Asia. If the US wanted to be a major world power, it needed a base in Asia like Britain, France, and even the Netherlands had.
Finally, people thought that the US would lose prestige and status. If the US let some other country take the Philippines, the US would not look like an important world power. The US would have looked like a country that was too weak or too timid to have an empire. This would have eroded the country’s standing in the world.
All of these are things that the US would arguably have lost if it had not annexed the Philippines in the early 1900s.