I would expect that the vast majority of those who do not pay it cannot afford it. Councils will use debt collection agencies. They do make the effort. However, this will have limited impact and councils also have to consider the ethical side of chasing money from some of those who are skint.
Over the years, one of the big issues with Council Tax has been the way Council Tax Support is funded. Before 2013, the Government covered the full cost of Council Tax Support. When that system was scrapped and replaced with local Council Tax Support schemes, the funding was cut and then rolled into general council budgets (this cost my local council c£30m per year). That meant councils had to pick up a lot more of the bill themselves. As money got tighter, many councils reduced how much support people could get or made the rules stricter. Unsurprisingly, that’s put more pressure on low‑income households and likely played a part in collection rates dropping in some areas.