Well, the "What if" question is always impossible to answer, but it seems clear from the prophecies (Ps. 22; Isa. 53) that crucifixion was always the plan. Crucifixion was "fitting" because of its cruelty, severity, and suffering. The image of crucifixion fits very well with what was happening there: Jesus identifying with the sufferings of humanity, bearing the sins of the world on himself, and experiencing the wrath of God in the place of all humankind. It fits the image of sacrifice. A slit throat, being set on fire, being killed with a sword could all be easily attributed to mere martyrdom, but theologically the Bible is firm that far more than martyrdom is what is happening at the cross. Therefore it's both a fitting image and a suitable process to represent the theology of redemption.

And it would have mattered to God and theology. The Old Testament model is that forgiveness comes through the shedding of blood. To have been strangled or set on fire wouldn't speak the same message.

Did the method by which the Romans executed Jesus have to be crucifixion? Jesus died from being beaten, hung on a cross and a spear to his side. What if the Romans had simply slit his throat or set him on fire or tortured him in some other medieval way? Would it matter to god or future followers? I know people say the way he died is the most painful method but surely you could make any method at least as painful/humiliating? Further still, if this were not the case would the cross symbol be integral to Christianity?