A severe asthma attack is an
emergency, not something a walk-in clinic can reliably handle. An asthma attack can range anywhere from moderate difficulty breathing (as though you're breathing through a straw or wet towel) to literal inability to breathe.
The treatment can be relatively simple, but in those cases you usually don't go to the ED to begin with: an in-home nebulizer device costs a few hundred dollars (uninsured) and pays for itself by saving only one ED trip (even if insured). The exception, naturally, is if your asthma is undiagnosed or usually not problematic, or if you run out of medication and need more in a hurry.
If it goes beyond that, the patient will often require intravenous medication, which a typical walk-in clinic is not equipped to administer. Even if the attack is obliging enough to occur when the clinic is open, all they will do is send you to the nearest emergency department.
The most severe attacks will require EMTs on-site, because the patient could easily die before reaching the hospital. One sometimes goes to the hospital in cases where one fears it will escalate to this as a simple preemptive, because if it goes badly you die.
It's true that there is a cost to handling all of this, and that needs to be understood. Still, it's worth remembering that this isn't some frivolous waste of resources: said resources are being applied to a legitimate need. The alternative is not to accept a lesser "level of service", it is to accept a nontrivial risk of dying of suffocation.