There's multiple kinds of olives, each producing different kinds of oil. The processing used also affects the end result (extra virgin olive oil comes from the first press, less quality oils from successive processing).
* Picual is the most popular one. It has a strong taste, that "bite" that you mention, and a long shelf life. This means that it is the most exported one. That is probably why you (and others) have associated olive oil with that strong profile. It is however not the only kind out there!
* Cornicabra is the second most popular one. It is milder than Picual in taste, but still strong and durable. You might find it in some specialist shops.
* Arbequina has less of that "bite" that you mention at the end. Unfortunately it has a shorter shelf life as well, which means that it also tends to be exported less.
Because of this is unlikely that you will find it outside of Spain. At the same time it would be the best kind for making mayonnaise, especially if you don't like the bite at the back of the throat (that some people actually enjoy).
And these are just the 3 more popular varieties. There's many more local, smaller ones.