Does excellent 4G coverage imply excellent 3G coverage? Is the primary benefit of 4G LTE speed, and not signal strength?

I want to sign up on T-Mobile with a Moto E, which is not an LTE enabled phone.  Of course 3G is passe, so 3G coverage isn't clearly indicated on the coverage map.  Do areas with 4G coverage always have 3G coverage, just slower?
Simple answer: No.

3G (also called UMTS) is a different techology than 4G (also called LTE).

Both technologies don't share any common base technology (starting with the frequencies and going over to the encoding technologies). So your carrier has two different nets, one with 3G and another with 4G. If you are lucky being near a 4G base station you have an excellent signal strength for 4G but the next 3G base station could be far away so that you have no signal at all.

It is not uncommon for carriers to have both technologies in one place but there is no provision for this.