1 amp isn't a trickle charger. Phones use to charge at 1 amp or less before we all became power hungry. Many cheap chargers nowadays still max out at 1 amp. Yes it would work, but as others pointed out, it'll charge much slower, if at all. It also may put more stress on the charger itself by being run right at maximum output all the time, and cause it to fail quicker. I don't know that this would happen for sure, but something to consider.
Should you worry about charge speeds on a phone? Not really. They have safeguards built in to limit the charge rates if needed. I.e. if it gets too hot. Batteries also have a "columb," or C rating. This is the amount of energy transfer compared to overall capacity. Li-ion batteries are safe to charge up to 1C, meaning the charge current can equal the overall capacity. For example, my LG V20 has a 3,400 mAh capacity. At 1C, I could charge it safely up to 3,400 mA, or 3.4A, without risking any damage. Even when quick charging, it maxes out the charge rate at about 3A, so it's safe to do. There are other Li-ion applications where the batteries can be charged much quicker, but they are specially designed to handle the higher C ratings and not something we deal with when it comes to charging our phones.
Heat is one of the biggest killers of the batteries, but even then they are generally good up to roughly 150°F. The other big killers are draining it completely and holding it at high charge levels over an extended period of time.
My usual advise is get the strongest charger available if possible. This not only ensures adequate power for your current device, but also gives more flexibility so you can use it longer and over multiple devices as the technology advances.