Why it matters

This one mostly answers itself once the arc is chosen, but it is worth laying out because the answer is less obvious than it looks. A three-hour train usually beats a one-hour flight door to door, and people consistently underestimate that.

A one-hour flight is never one hour. It is transit to an out-of-town airport, a bag drop, a security queue, a boarding process, the flight, the wait at the carousel, and transit from another out-of-town airport. Realistically four to five hours, and a broken day either side of it.

A high-speed train is city center to city center, with a ten-minute arrival buffer and no meaningful security.

The options

This is not really a preference — it is a matter of which routes have usable rail.

Take the train

Route	Time
London → Paris	2h20
Paris → Amsterdam	3h20
Rome → Naples	~1h10
Rome → Florence	~1h30
Madrid → Barcelona	~2h30
Madrid → Seville	~2h30
Seville → Granada	~2h30

All city center to city center. All comfortably better than flying.

Fly — no practical rail

Route	Time
Athens → anywhere	2h+
Rome → Barcelona	~2h
Barcelona → Paris	~2h (or ~6h30 by train)

Greece has no useful rail connection to the rest of Europe, and the Italy–Spain crossing is a sea crossing. Those are flights, and no amount of preference changes it.

Barcelona → Paris is the one genuine judgment call: 6h30 by direct train versus about 2 hours in the air. The train is roughly a wash on door-to-door time and considerably more pleasant; the flight wins if the day is tight.

My recommendation

Trains wherever the route offers them; flights only for the sea crossings. This is close to a non-decision.

One practical warning that is worth more than the routing advice: budget carriers have strict and aggressively enforced bag rules. With four people and two weeks of luggage, the add-on charges and the risk of a gate-side repack make them a poorer deal than the headline fare suggests. Full-service carriers or trains are usually the better call for a family this size, even before comfort enters into it.

What would change my mind

Very little — this follows from Which arc? rather than standing on its own. The northern arc is almost entirely trains, which is one of its strongest practical arguments. The Mediterranean arc is almost entirely flights, which is one of its real costs.