You've clearly misunderstood what's happening with pain management in Europe.
Firstly, it has nothing to do with cost. European pain management is expensive but available.
1) If you are in short term pain you will get pain relief. A wide range of options wil be available, from non-pharmacological options at one end (easing fear, for example) through non-opioids (paracetamol, ibuprofen, entonox) but using opioids when needed. Here's a guide form one of the English NHS Ambo trusts: https://www.swast.nhs.uk/Downloads/Clinical%20Guidelines%20S... You should be reassured that the entire document talks about properly treating pain, and reducing the pain the patient is in. There is no mention of cost in that document.
2) The reason they don't just go direct to opioids is because opioids have side effects. They have more side effects than other pain control meds. This causes harm to the patient.
3) If you are in long term pain things are more complex. People in long term pain fall into different groups. Some of them will be cured from their long term pain with lifestyle modification. Most, but not all, people with lower back pain will be cured if they lose weight and get exercise. It would be cheaper to give them opioids, but because diet and exercise are more effective and safer they get a package of care that includes access to a gym and physiotherapist, access to weight loss clubs, access to dieticians. Some people with long term pain will never be cured, but will be able to manage with access to a pain clinic. But some people with long term pain will always have that pain. There is no cure for it. We can't just give these people all the opioids they want because they develop a tolerance, and need an increased dose. At some point you have people taking very high -dangerous- levels of opioids but still not getting pain relief. These people need a package of bio-psycho-social care that includes access to specialist pain management clinics. Again, this is more expensive than just giving people opioids.