When connecting via a cellular data network, all data will be routed via your carrier's Access Point Name (https://en.wikipedia.org/wiki/Access_Point_Name).
Most carriers only have servers in their home region capable of handling the traffic. It is possible to have regional servers for this, but most carriers don't bother with the expense. Roaming data is usually expensive or restricted in usage - so there's traditionally been very little demand for higher speed connectivity.
Travel eSIMs are usually just a regular SIM with a very limited plan from a carrier that has favourable roaming deals.
No, that's never really been the case. It's technically possible (and called "local breakout"), but for various reasons I also don't fully understand, it's usually not done that way.
One is legal liability for user actions; another is accessibility of services at home, such as banking apps, that are probably more comfortable with a familiar IP.
Partly I think it's skirting around local regulations (e.g. if a country required SIMs to be registered with the owner's information, but you provision the SIM from a nearby country's telco and route traffic through there maybe you can get around it).
But as far as I understand this has nothing to do with eSIMs and a lot to do with Airalo trying to cover every corner of the world for cheap.
The host roaming network is just selling in bulk, but does not own the customer.