LeaLeander nails it.
The business model for airlines has changed massively. If you select solely on price then expect to pay for extras. What we have found is that very often adding up the luggage, hand luggage, seat allocation, checkin, credit card, meal, you name it up and makes very little difference between budget and full service carriers. Which when you think about it makes sense - if you want full service, then pay. If that is with squeezy then there you go.
With that in mind - we check the routes against airlines like BA, etc when booking holidays and have actually found them cheaper sometimes and whilst have never booked a seat in advance have always worked things out at check in. (always online - always when it opens) a party of 5 is hard to sit together when it is narrow body and never much fun in a wide body.
Airline pricing is a dark art. They have to shift the maximum of product for the maximum price and cater for last minute changes, cancellations and flexible ticket holders. Every dollar you shave off the price, the airline takes an option away.
A fully flexible economy ticket to NYC frequently costs a lot more than a business class ticket. But then you can change your flight to the next one upto an hour before departure...
If it matters to you that much - then pay. I recognise that in the OPs case they have scraped together to pay the flight. If you cant afford to pay to sit together, then you are taking a chance that it can be sorted at check in. if that risk is not acceptable then either suck it up or cough up.
However - to claim that the person who has paid should then willingly give it up so that you should get your desire for nothing smacks of a level of entitlement that beggars belief. Perhaps I should let you have the lunch i paid for on the flight? The one you didnt pay for?