Perhaps sacrifice is letting go of material attachments? Camel through the eye of the needle? Fasting and prayer, to be open to angels?
Yes, all of that could figure in. Sacrifice was/is on a continuum and many varieties are acceptable. Some were not even blood sacrifices but were acceptable and never turned away as long as everything was done according to divine plan.
However it remains that where the need is great, the sacrifice must match it. These greater sacrifices were blood sacrifices, with the remains being burned without the camp (where all hope is abandoned and one is truly alone, or so it would seem).
Just my opinion here, but it is as if God gets this type of sacrifice in such a position as to make all help impossible, unless it comes from Him. Earthly hope dies along with all attachments (according to what the one giving it is capable of receiving). It is here in this helpless and weak condition that the Divine agency acts upon the sacrifice, consuming it with fire.
Through progressive revelation we now know that life is returned to the one giving the sacrifice (resurrection) in a new and better form, not capable of corruption. This is the sacrifice Elijah had confidence in, and experienced (in a figure).
This type of sacrifice is not for the faint of heart and of course must be by divine appointment. It is a step by step process and carefully done, it cannot be rushed. When it all comes together, the very heavenlies are moved in a way never seen before.