TL;DR – I only connect on LinkedIn with people who I could personally introduce and recommend.
I realize that many people use LinkedIn as tool to grow their professional networks, and to “bookmark” people they have met at conferences etc.
A core function of LinkedIn is the ability to connect to people two or three degrees away through introductions. All my connections are visible to people several degrees away as someone they could get an introduction to. So, to me, it only makes sense to connect to people I feel confident personally and professionally recommending to someone else.
I am more than happy when anyone reaches out to me through LinkedIn, because I love to help out however I can. I am just highly likely to move the conversation to email.