Invasions are supposed to count, but they don't stack with arena (so if you win 20 invasions and 30 arena matches, you don't rank up)
I think the lack of reds is due to these problems:
1) The invadee pool is as big as the playerbase itself (thanks to hollows being able to get invaded and invasions happening after the bosses are killed), while the invaders are more restricted due to the lack of a reusable Red Eye Orb. This spreads invaders too thin.
2) Since you can't heal in invasions and the matchmaking system prioritizes groups of players, you end up with extremely unfair battles, which are much harder to win than in Dark Souls 1. You really need some sort of spellcasting and a Heal spell. Basically, you need some sort of "invader build". This makes invasions really annoying for a lot of builds. Arena doesn't have this problem, sure, some builds are worse than others and beating a min maxed buff build with a sub-optimal build is hard, but it's nowhere near as hard as beating 3 people with melee only while they spam spells all over the place, and on top of this, you can't heal.
3) Arena is actually really fun, vastly improved over Dark Souls 1, and it's completely free for reds. Since arena is mandatory for invaders (in order to farm the orbs) I suspect many will do what I did - discover how much better arena is and simply keep doing arena and neglecting invasions.
I suspect it's mainly reason 1, though. In Dark Souls 1, the vast majority of people were hollows, and only a smaller group was human and thus eligible for invasions. It was also harder to stay human, which kept the invadee pool more restricted.
It's interesting that the change many people bitched about, the ability to invade hollows and people who have already beaten the boss had the opposite effect we thought it would have - it made invasions extremely rare.