Beware of lots of scammy websites masquerading as something official. At least one of these is giving an 0871 number. If the website isn't .gov.uk then it's not official and may be a nasty scam.

As far as I can see, the 0300 number is the only official contact. The poor service is just another symptom of badly run government departments.

While 03 numbers are charged the same as 01 and 02 numbers, there may be problems when called from abroad as foreign phone systems may not properly recognise them.

As you have discovered, the phone lines for many so-called public services are not there to actually help you. They are trying to make you go away and keep quiet. This is nothing to do with 0300 or any other sort of phone number. My local council has outsourced itself to C*apita. I've not tried to phone them recently but I've heard that getting through to somebody who can help is now very difficult.

Before you visit in person, make sure they can actually take visitors in person. Should be on the website.

I wish you luck.