Yeah I bet that is EXACTLY what happened!
Even if you dont drive fast, there may come a time where you have to brake hard or swerve to avoid an accident and thats when you will really depend on your tyres
As an alternative to cheapos look for some partworns. PM gixerboy or look on ebay etc. Wont last as long sure but at least they will be decent.
my advice is, fit the best tyre you can afford, and get them fitted by a PROPER tyre fitting business... like armstrong beattie, phillip white etc, one where people are actually trained to fit tyres, or your budget buy will turn into a nightmare if a rim is damaged.
i seriously wouldnt recommend anyone fitting triangles if they value their life
All cars in our family use Lassa from Central Tyres, and have never had any issues with grip etc.