I don't buy the less is more idea. Copying this post I wrote a little while back:
I think it's hard to put a number on as it depends on your availability to write applications. If you had absolutely nothing to do bar writing applications, I'd say write as many as you can (15+). Not everyone is in that position, however, especially if you can only write applications in the evening/on weekends. In that case, 5-10 is a good amount.
Push yourself to write applications as you can. As much as I give advice about maximising the strength of your applications, it is still very challenging to convert an application to an interview and then an interview to a training contract, so by applying to more firms, you significantly increase your likelihood of securing a training contract. That said, I also think it's important to carve out time for yourself and give yourself lots of breaks. Many candidates won't secure a training contract in the first cycle, in which case, those who work themselves too hard in one application cycle risk burning themselves out too quickly.
I try not to treat it as a trade-off between quantity and quality. Applications should always be of a very high standard and it's rarely worth sacrificing that (unless it's genuinely taking you weeks for every application or it's close to the deadline). There is only one candidate I can think of who writes applications to such a high standard (but to the detriment of time) that I tell him he needs to force himself to spend less time on quality and more on quantity.
In terms of why you're getting rejected at the application stage, what are your grades like? Do you have any existing legal experience?
