The Meeadowbank Velodrome hasn't shut
: it was under threat but for the moment its management has been taken over by Edinbrugh Road Club. They were even offering free taster sessions in June edfoc.org.uk/events/event/meadowbank-velodrome-taster-sessions/
The disadvantage it has is that it is not a covered velodrome and is therefore weather dependent.
In terms of accessibility, there are lots of ways that people can learn track cycling inexpensively. Ds' cycling club gets kids started (on club bikes) on a slightly banked outside circuit - there are quite a few of them around the country. The membership isn't expensive: max of £25/term and that includes the coaching.
Sessions in the Sir Chris Hoy velodrome are more expensive (once you're accredited, they're c£5/hour - a bit more if you have to hire a bike and shoes) - but they also do taster sessions. You can "hire" a bike from the club for, iirc, £2/week.
Ds' school took kids to the velodrome to give it a go (unfortunately, ds didn't get to go as he was already full accredited
)
Rugby where we are (Glasgow) is also not elitist. 95% of the kids at the local club are from local state schools - and there are very few posh cars in the car park! It costs the princely sum of £65/year for child or £110 for a family (and even then. if people have an issue with affording it, we'll come to an arrangement) - which, by all accounts, is a lot less than the local football clubs. The only other unavoidable expense is a gumshield, which can be bought for less than £5. There is a boot and kit bag where people return grown-out-of-stuff so that people can check there before buying new stuff (new boots cost c£30). Game kit is paid for by the club and washed by whichever muggins volunteers
Most sports really do try hard to make themselves accessible.