The expensive/branded ones don't set your house on fire
You might assume that would be true but it really isn't. Despite the much smaller market penetration of high cost high end ebikes there has been a fair few issues and recalls, here is one below;
We take a closer look at one of the most famous electric bike companies, Specialized, and find out why its products are loved by cyclists all over the world.
electricbikeaction.com
When you look at the average price of bikes sold including ebikes its only around £400-600 and doesn't allow for many high end ebikes overall in the market place as a percentage of ebikes sold yet you aren't seeing something like 200 cheaper ebike fires for every high end ebike fire which is what you would need for the same level of risk. Halfords have something like 40% of the UK bike market by volume, around 25p in every £1 spent on cycling in the UK is spent in Halfords and yet they do many low cost ebike models without any fire risk recalls so far I think. Last statistic I saw was 11% of all of their bike sales were ebikes. That's a huge number of ebikes out there many of which have retailed sub £1000.
Of course Halfords ebikes are full certified to sell in the UK and go through the correct testing.
I feel the real safety risk is from personally imported batteries i.e. aliexpress which aren't correctly certified, personally imported ebikes, e-scooters & hoverboards (which for some reason get mixed in with ebikes in news reports) and of course home ebike conversions. Then of course there is user error which seems to make up a lot of fires anyway and can affect any ebike of any price point. I actually feel there is less fire risk with many of the cheaper fully certified ebikes simply because they aren't stressing the cells of the battery like mid-drive motors which have very high current demands when hill climbing.
I really don't think it's as simple as price in fact when you look at standard bicycles there are much greater safety issues with more expensive performance bicycles. Often these are made to be lightweight so you end up with carbon fibre which has been a huge cause of product recalls. It's hard to think of a single manufacturer that hasn't had a carbon fibre fork recall. Making products to be lightweight often reduces their strength and durability, lowers weight limits and if you read a Instruction Manual from brands like Giant they actually give warnings about being a weaker and shorter life product for their performance bicycles.