Can I get the same benefit of a mesh Wi-Fi system by setting my different routers to AP mode and using the same name and password?

Yes, you never needed mesh to do a roaming network. All APs support this, because there's nothing special for the APs to do. The 802.11 (Wi-Fi) standard leaves all roaming decisions up to the clients, and clients are supposed to pick the best AP for their needs, as long as it is publishing the network name (SSID) that the client wants to join.

You never needed mesh to do wireless backhauls between APs. Not all APs support this, but there were APs that supported this long before mesh systems hit the market.

You never even needed mesh to do a self-assembling/self-optimizing tree structure of wireless backhauls between APs. Not all APs support this, but there were APs that supported this long before mesh APs hit the market.

You never needed mesh to have APs that play tricks on clients to try to make the clients roam better (I say "tricks", because the 802.11 (Wi-Fi) standard leaves all roaming decisions up to the clients; it does not provide a mechanism for an AP to tell a client how to roam). Not all APs support this, but there were APs that supported this long before mesh APs hit the market.

Mesh is short for "mesh topology", which is where any AP can talk directly wirelessly to any other AP in range, not just to its designated upstream or downstream APs in a tree structure, but cross-linked like a web. That's really all that mesh adds. Mesh systems usually have all the other things listed above, but those aren't what makes them mesh. Only the cross-linked topology is what makes them mesh.