Hi. First of all, noone can really say that they will 'easily' embark on a career at the Bar. The process is difficult for everyone. You only have to look at the statistics to see that there are far too many applicants for pupillage than there are places for them. Not only that, but the recent pandemic has had a terrible effect on many chambers. Significant numbers of pupillage places are being deferred and the competition for next year is likely to grow even greater with a large proportion of chambers deciding that they cannot support more pupils.
That being said, it is good to be aware of the mountain ahead so that you can focus on how best to climb it. The Bar is far more diverse nowadays in where the newest members hail from. A good application is not 'just' about academics, but academics are a significant tickbox which must be checked alongside other factors. Generally speaking, a first class degree is likely to satisfy that tickbox at most sets outside the top flight sets that practice commercial/specialist civil areas of law. The name of your university is not likely to be a significant issue unless it has an obvious reputation for low quality. You are usually required to set out your academic record at least to A-level (sometimes to GCSE), so that those making the decisions can check for a general academic pattern. If you have a very poor record prior to university and then a great degree you are likely to be questioned about that. You will need a convincing explanation that does not seek to simply shift the blame on something else. Some of the best candidates I have seen had poor academic records prior to university, but were able to explain - essentially - how they had 'grown up' and developed their full potential. Recognising your own weaknesses and improving/countering them are an important part of the advocate's toolset, so demonstrating it within your application is not a bad thing at all.
When looking through forms we are also concerned about people who are 'just' academic and do not have a rounded character. The point I make time and time again to hopefuls is that it is massively important to demonstrate you are an 'interesting' person. You should engage in activites and skills that demonstrate your ability to interact with different levels of society, your competitive streak, your love of life etc. I have seen forms rejected because a candidate, whilst having a solid academic record, simply did nothing else but study. There are too many candidates for places to afford not to catch your assessor's eye. In turn, the lower your academic record, the more you must compensate for it in other areas of your application - whether it be in legal work experience, activites, competitions etc.
A good idea is to look at the chambers that interest you, or a range of chambers that are in your chosen practice/geographical area, and read the profiles of their current pupils and recent tenants (up to about 5 years post-pupillage). Their profiles will usually give you a good indication of what their CV/application looked like. You should compare their acheivements and history to your own. Would you have been able to compete with them, had you applied alongside each other? That's a good starting point to assess your own application/background and see what you can do to improve.