I'm not sure companies can check birth certificates on the computer - I think there are data protection issues there - how comfortable would you feel about anyone being able to log and look at your personal data. also think about your own security - you only need your birth cert details to open a bank account these days; if someone could log onto that computer system and take those, you could have an account opened in your name which could lead to all sorts of problems. I also don't understand what you mean about something being "blatantly obvious".
Anyway, all employers and employment third parties require anyone joining them to prove who they are and that they have rights to work in the UK. A passport or birth certificate will do that as it prooves you are either a UK/EU citizen or that you have a valid visa to work in the UK. A driving licence is proof of ID but not proof of right to work.
So if you have a passport that should do, If you don't, then you need your birth cert. Why it has to be A4 size as opposed to the smaller version normally issued in the first instance I don't know.
I also don't know where your �60 came from - I just googled and came up with the following
http://www.ukcertificates.com/birth_cert.php
which says its �19.95 for standard deliervy service - admittedly it's �45 for a speedier service but still not �60.