I applied a year ago, David. The Thai London embassy emailed me saying I should ignore the requirement for life insurance, as no longer needed. Too late, as I'd already bought it for £1,100!
If you are clear the landlord is responsible your last resort is to call a plumber, pay to get it fixed and deduct the cost from your rent. Landlord will likely deduct it from your deposit when you leave, but there's no guarantee you'd get that back anyway.
It seems so inconsistent though, Annie. If the AML legislation required you to show a tax return, why didn't it require me to show one? From many reports I've heard, if a foreigner turns up at a bank to open an account, the bank puts pressure on them to buy insurance as part of the deal, or turns them down. Their reticence to open an account is overcome by a little personal incentive, which is why agents charge 5,000. But if the foreigner goes to Immigration first, the bank understands they probably know the procedure from a government agency, and can't be blackmailed allegedly!
If you don't want to use an agent, it is possible to do it on your own. Make sure the place you're staying has registered your stay with Immigration. You need a long term lease/contract to show it's a relatively permanent address. Take these to Immigration and ask for a Letter of Residency, which most banks require these days. You'll need photocopies of your passport bio page, entry stamp and visa. And 2x passport size photos. Then take that letter to a bank and say you want to open an account. I did this in Jomtien on Monday. It took all day!
Former HR guy from UK here. We used them for senior positions only, like owners of businesses a client had bought had non-compete clauses and, given the cost of buying the business, no one could complain they had to work to live. But we absolutely knew we could not enforce that on employees.
Maybe a more immediate issue here is whether you can stay in Thailand while a new work permit is sorted out. If you have left the Bangkok job, your work permit and visa are no longer valid and you ought to leave Thailand within a day or two. Then your new employer in Phuket should support your application for a new work permit and visa