I don't understand why the Roam won't charge with a 5V/1A power supply. 
I know Sonos says it doesn't have enough power but that is not a sufficient answer. 
as long as a charger has a higher Voltage capacity than the battery, the battery can be charged. 
The Amp rating of the Power Supply just determines how fast the battery will be charged. 
For Example: 

If I use a 5V/1 amp power supply to charge the Roam with a 5000mAh battery and required charge voltage of 4.2V

5Vx1amp=5W

5W/4.2V=1.19Amp

5000mAh/1.19Amp=4.2 Hours (Time required to Charge the Roam)

So I understand that a 1Amp charger will take long but not charge the battery doesn't make sense. 
 

My only conclusion is that Sonos programmed the built in Charger in the Roam that it will only allow charge if the charger receives more than 1 amp of current. 

I can understand this out of a User friendly perspective to not allow chargers that will be slow so that people who can't read a label or do Math not get frustrated with a slow charge. 
 

But why not allow the slow charge because it is much better for battery health because it will generate much less heat. 
 

My speaker gets so hot with a 2A charger because of the high current.

Just a tip Sonos:

Look at what apple does with their iPad battery configurations. They put the batteries in serie. By doing this they get a few advantages:

1)Same charge time but at half the current which is amazing for battery health. 

2)Running higher voltages means all wires can be thinner which saves a lot of Copper. 
3)Higher voltage can drive equipment with higher power requirement. (Again, with less heat)