Is it safer to use a power supply that isn't high amps?
There are lots of variables at play, but in general, it's never a good idea to operate a small load from a huge supply unless the small load has safety features designed to protect against what a huge supply could do under abnormal conditions - appropriate fusing certainly comes to mind.

Imagine your load fails and presents a short-circuit to the source.

A 1A current-limited power supply will deliver some percentage above its 1A rating - most manufacturers target overload thresholds between 110 and 125 percent of nominal to allow some margin. So, your load can potentially suck 1.25A unless something interrupts the current.

Imagine now you have a 10A supply. That 1.25A is now potentially 12.5A, and now if your downstream device doesn't have some integral protection, you're in trouble.

Imagine you now have a power supply that doesn't current limit, but instead limits the power going to the load. The amps will increase but the volts will decrease. A 5V, 1A power supply could become a 1V, 5A supply, or a 0.5V, 10A supply. Now imagine a 5V 10A supply which behaves this way.

Also, fuses tend to work better when there's sufficient current available to clear them, and may not work at all if the source isn't stiff enough. A 10A supply should easily clear a 1A fuse under abnormals - a 1A supply would likely deliver current indefinitely into a short through a 1A fuse.