But here's a site with the data. The regular heartbeat picture has not been standard over EONIA's history. See the crises of 2001 and 2008 (from another site because larger):

For 2009:

It could be read as injection of liquidity then careful tending of same from mid-2009... Followed in mid-2010 by sterilisation having, as you suggest, a net draining effect. (Though I suppose someone else might see a different pattern, turtles etc).

There's a similar stable heartbeat pattern in mid-decade, presumably between higher upper and lower bounds?