Generally speaking, you need a reflexive pronoun of some type if not having one causes ambiguity in who did what to whom. Assume English had no reflexives. "I tossed the ball to I" and "You tossed the ball to you" (assuming singular "you") are quite understandable as one person tossing the ball to themselves, thus no reflexive needed. "She tossed the ball to he", acceptable. But suppose you had "She tossed the ball to she", and it's been previously established there is more than one woman present. Did person A throw the ball to Person B, or did she toss the ball up in the air and caught it herself? There's no way to tell. An example like that is why German only has a distinctive reflexive in third person; first and second person don't have that same ambiguity.
You can get rid of the reflexive entirely if your verb system, to use one possibility, allows a given verb to indicate if it takes a direct or indirect object, or if it does not. Let's say that English verbs used the affix a- to indicate the verb is ditransitive (it has both a direct and indirect object) and no affix to indicate purely transitive (it only has a direct object).
"Mary and Susan were on the field. Mary had the ball. She a-threw the ball to she." - Mary threw the ball to Susan.
"Mary and Susan were on the field. Mary had the ball. She threw the ball to she." - Mary threw the ball up in the air and caught it herself.
Under a system like that, English wouldn't need a reflexive pronoun because the verb would make it clear whether the subject was performing the action on themselves, or using themselves as the indirect object, or whether something else was the recipient of the action. One could argue the affix a- is similar to the single Icelandic reflexive pronoun that's used for everything.
In Inuktitut, verbs that are are equivalent to English intransitive verbs use one suffix attached to them to indicate the subject:
pisuk to walk
pisuktunga I am walking
pisuktutit You (singular) are walking
pisuktuq He/She/It is walking
Specific verbs, equivalent to transitive verbs, use a different set of suffixes that indicate both subject and object:
qukiq to shoot something
qukiqtagit I shoot you
qukiqtara I shoot him/her/it
qukiqtarma You shoot me
qukiqtait You shoot him/her/it
qukiqtaanga He/she/it shoot me
qukiqtaatit He/she/it shoot you
qukiqtanga He/she/it shoot that other he/she/it
Here's where the reflexives come in: if you don't use the second set of suffixes, but instead use the first set, then the verb becomes a reflexive: the subject did something to themselves.
qukiqtunga I shoot myself
qukiqtutit You shoot yourself
qukiqtuq He/She/It shoot themself
Attaching the suffix that goes on intransitive verbs onto a transitive verb turns that verb into a reflexive.
So, if your verb conjugation allows that sort of thing, then you can get away with not using reflexive pronouns. The verb indicates it.
On the other hand, if your verb doesn't allow you to do that, then you need reflexives in order to remove the ambiguity when you get into situations where it's not clear who is doing what to whom.
Another way: go back to that first English example I used. Suppose English had multiple third person singular pronouns, such that you didn't just have "she", but "she-1", "she-2", and perhaps more. Then what does the example look like?
"Mary and Susan were on the field. Mary had the ball. She-1 threw the ball to she-2." - Mary threw the ball to Susan.
"Mary and Susan were on the field. Mary had the ball. She-1 threw the ball to she-1." - Mary threw the ball up in the air and caught it herself.
No ambiguity. The context makes it clear that Mary is she-1 (Mary had the ball, she-1 threw the ball, therefore Mary is she-1, so Susan is she-2).
A third option is context. You might have a language where you do need to gather more information in order to determine what's going on.
"Mary and Susan were on the field. Mary had the ball. She threw the ball to she. Susan threw it back." - Mary obviously threw the ball to Susan.
"Mary and Susan were on the field. Mary had the ball. She threw the ball to she. Susan then asked for the ball." - Mary apparently didn't throw the ball to Susan, since Susan asked for it after it had been thrown and caught.
So, three possible ways to get away without reflexive pronouns. So it's not a matter of how common or uncommon it is, but does the grammar of the language allow you to get away without them, or is it necessary because there's no other way of providing the same information?