The issue is that £2700 may not be high if you have 3 teens at home, but that also applies to anyone who actually earns this amount, or less, without being topped up to it. Up until recently I earned less, as a single mother, too, but because my net income was still above the threshold I wouldn't have been able to claim anything other than child benefit. I now earn the same as what you get a month, but I work 60-80 hours every week to keep up with the workload.
So there will be a whole bunch of people in full-time work, earning less, not entitled to the same amount of money.... and that seems somewhat wrong. Because once you're above the threshold it doesn't matter what your circumstances are, how many children you have, whether you own or rent, whether one of your children has a disability.
And then resentment builds, because what is the point in working in a job, often promoted into it with a significantly higher workload, full time, if people can work part-time or in NMW jobs and they end up on a higher income?
The whole system seems somewhat screwed up.
And, OP, quite honestly, to then bring in the question of savings seems a tad ignorant when we are in a situation where many people on less than you can barely afford to live, let alone save significant amounts.
I don't blame you for the state of things in this country, of course, but many people on an average wage would look at your single income and be green with envy.