It's a very painful process, I completely agree with you. For much of the last 2 years, I've felt completely the same way. Rejection after rejection, from firms of every rank. The LinkedIn element makes it even worse - pretty much everyone I knew also at Oxford was going on to become an investment banker/corporate lawyer/management consultant - and there I was, with no graduate job, feeling a complete failure. BA Oxon & a bartender - I got more than my fair share of jokes about that from my pub colleagues.
The only thing I can really say is to keep plugging away at it - but to refine your efforts. I got 47 rejections over the last 3 years, and each time I tweaked my approach slightly. On time number 48, it worked, I got my first and only invite to a vac scheme interview, a vac scheme offer, a TC interview, and finally a TC, all at
Linklaters.
On the VS, I was constantly worried that I was too old (almost everyone on it was still a student), at a disadvantage because of my non-law degree, at a disadvantage due to my lack of commercial experience (most of the older vac schemers had already worked at banks or law firms), at a disadvantage due to having no other vac schemes to fall back on. None of those things actually mattered - by sheer hard graft on the work tasks, I was able to prove myself.
I know it sounds a bit lazy of me to say, but you really have just got to keep trying, and to rise above your inner doubts if you can. LinkedIn, like all social media, only shows one side of professional success, and not the struggles or issues the person had to overcome. There are very, very few people in life to whom everything comes naturally and easily, contrary to what social media would have you believe.
Some people naturally are good at the application process, or have lots of friends/family already in corporate law firms who can give them advice. I wasn't one of them, and by the sounds of it, nor are you. That makes it tougher, I won't lie, but it is still very much achievable, as long as you stay motivated and committed.