Motorizer - Yes it is common practice to depart with no provision for holding. Legally we are not required to do so and in reality this is a purely a business decision, it is not a safety issue. The price of doing so is the occasional diversion because this is a trade-off. This is amazing when you look at the costs involved. Firstly, there are additional landing and handling fees, security charges, parking costs etc. To that the costs of taxis and buses will have to be added. Then the additional crew costs have to be added in. If this crew would have to rest for at least the same amount of time they were on duty and that would almost certainly restrict what they could do next. The we have the fact that the aircraft is now in the wrong place. Finally, we have the passenger costs. Diverting for the want of 20 minutes holding fuel sends the wrong message regarding the integrity of of your future holiday. Not surprisingly, most people don't like diverting and with modern social media, the world and his dog knows what has happened. This has a price as well.
Regarding the why you can't ask for a "straight in" approach question; that's simple. Most other operators would be in a similar position and only if you declare an emergency.
I also note your comments regarding unwell passengers. Passengers are the most unreliable things on modern aircraft and probably do as much damage to airline's schedules and bottom lines as slot delays. They are probably responsible for more diversions than poor weather. They regularly become unwell when things start going wrong and the effect of doing so causes untold damage to the operations of both airports and airlines. What's really galling is that when dealing with these passengers you often find that they have pretty dreadful medical backgrounds and pre-existing medical conditions and they don't mind inconveniencing hundreds of other passengers when their condition worsens under stress of travel. Maybe it's about time we asked for a contribution towards the costs they force all of us to incur?
The good news is that it appears you were flying with a "proper" operator and a good crew. One that employs and trains its personnel to do the correct thing and not bend or break the rules. It's also one that appears to do its best to get you to where you bought a ticket for, even when things don't go according to plan.