>You don't get government unemployment benefits in the UK if you have >16k in savings
This isn't true. You'll get new style JSA for 6 months [1] pending you have contributed NI for the past 2/3 years which almost all will have.
https://www.gov.uk/jobseekers-allowance/eligibility
You're talking about universal credit which yes you won't qualify for with greater than 16k savings (pension savings or HTB ISA are however are excluded).