Yes, gamblers should take full responsibility just like shopaholics, alcoholics and other people with addictions. If we start dividing this responsibility between those who advertise (of their product which, when used in reasonable quantities, does no harm) with people who consume it uncontrollably and face unpleasant consequences, we will harm all parties.
Same in cigarette advertisements - "smoking is dangerous to your health"
Same in liquor advertisements - "drink moderately"
Some gambling sites also have a banner or terms pointing to "gamble moderately" or "responsible gambling". Since these companies are promoting their products and services, obviously they won't say stop doing this or that.
We, as a consumer and clients, should be responsible for what we are doing.