Here in SE13 I'd say one of the major barriers is the lack of support by school leadership for music as a curriculum subject.
My own experience as a primary school teacher is that there is so much pressure on children making progress in Literacy and numeracy that that lots of class teachers panic when children disappear out of parts of lessons for 30 minutes once or twice a week. Personally, it's not something that bothers me as I spend at least 5 hours a week teaching Literacy and numeracy - if a child doesn't 'get' what we're doing in 4.5 hours, I doubt that the 30 minutes that they are in a recorder/brass/woodwind lesson will be the dealbreaker.
I am completely biased as I'm someone who played musical instruments all the way through primary and secondary school, loved singing in choirs, playing in the youth orchestra etc. and I would like every child I teach to have the opportunity to do the same. Sadly, until primary schools are no longer judged by their KS2 test results (by Ofsted, the press and by parents), I expect that most Headteachers will feel they are justified in claiming that parents want high KS2 results and so music has to be an extra that is squeezed in here and there. It's not the case in all schools - I know of a couple of primaries near here who have made music a priority. At those schools every single child will learn at least one instrument throughout their time at primary school.
Another possible barrier is the apparent lack of opportunity to perform through the school and the age limits put on other opportunities. For example, DD1 took up the trumpet aged 7.5 (because she had been desperate to for ages, had picked up the recorder/reading music quickly, had enough adult teeth and there were spaces in the brass teacher's timetable) having a 30 minute group lesson, at lunchtimes, once a week. In Y3 she was put in a group with 2 children, one from Y5, one from Y6 and took grade 2 after a year of playing (she passed with distinction). The other two were able to go to a very local schools event for brass and woodwind players along with all other Y5+ pupils who were having school music lessons. DD1 wasn't allowed to go because she was in Y3. Unbeknown to me, her brass teacher and the recorder teacher had put her name forward to attend the event. Being in Y3 was the only stumbling block. So, a highly motivated, keen and musical pupil missed out because she's not in the correct year group. Other children from DD1's school returned from the event saying, "you should have been there - X can only play 3 notes and she came, you'd have loved it". She won't be in the correct year group to participate in any of the Lewisham/Southwark/Lambeth music events for another year from now. I'm not sure that's a great way to enthuse young musicians! Luckily for DD1, she does get to play at church once a month and through the Music Service Saturday morning groups so it's not as though being barred from that event (which was for the launch of a new group) has prevented her from playing. Chances are, there will be children in Y4 for whom that would have been their only easy opportunity to engage with music socially.