Lack of a secret ballot is a greater problem
I complained to the Electoral Commission a couple of years ago that all ballot papers in the UK are numbered and that that number is recorded by the polling clerk against your name. That means that a person's vote is not secret and so it's not a secret ballot as is standard in proper democracies.
They said it is done to tackle voter fraud and that I should just trust the government not to abuse it. I retorted that it is ridiculous that anyone can rock up to the polling booth without ID or even the polling card and claim to be anyone they have a name and local address of and vote on their behalf. Until a couple of years ago I lived across from my polling station in a building with a shared letterbox. I could have easily popped over in the morning and vote as myself, pop over around lunch time claiming to be my next door neighbour and pop back around dinner time claiming to be another neighbour. A polling clerk who sees hundreds of people a day would not notice a thing. How can someone claim to have numbered ballots to protect against voter fraud if anyone could vote as anyone?
If requiring ID (or at a minimum handing in the actual polling card!) means that they can introduce a secret ballot I am all for it.