I’m a curmudgeon regarding unsolicited sales inquiries through LinkedIn, which is why I only check it every few months. I’m constantly getting requests for a phone call to introduce me to some new product or service. 
This probably will not do any good, but I added this statement to the heading in my profile: “NOTE: I do not respond to sales solicitations.”
I don’t have the same problem as you on LinkedIn but you can turn InMail off (which I believe is the name they give to messaged from unconnected people):
Thanks, I’ll check it out.
I’m disappointed but unsurprised to hear that this is happening on LinkedIn. I was on there for many years but never liked it much, partly because LinkedIn did such a bad job with their apps and partly because they gradually deprecated the few things that I genuinely likes such as Groups. I’ve thought about going back, but honestly I don’t feel the need to be ‘connected’ in that way.
I don’t have the need either per se but because of speaking engagements and articles, I find that people try to find me, so I use it more as information. I seldom engage on the site except to occasionally post an article. I have little use for any form of social media, especially one filled with sales solicitations. 
I have people regularly connecting then soliciting immediately. If that happens, I immediately disconnect again.
I think it’s rude to do that without even opening a dialogue or building a relationship.
I had to delete my LinkedIn profile because the calls and messages were getting to be really egregious. My contact info is still on some lists though so my new favourite response is to say I’m now retired. It stops the calls in 30 seconds flat.
My rule of thumb with LinkedIn has always been: Do I actually know this person – have I ever talked, emailed or otherwise interacted with them before their LinkedIn communication? If not, I don’t connect, unless I have a definite reason that I want to get to know them.
Their initial message might give me that reason – but it’s incredibly rare.
I’ve barely paid it heed in the last 5 years or so, but I had even stricter rules. As the original intent was about professional relationships, and a lot of that was about job hunting, my rule was “if someone called me about this person and asked if I should hire them, would I have anything meaningful to say?” In other words, unless I had worked with the person enough to actually know them (and like them), then they were not getting connected.
I’ve gone back and forth on this. My initial feeling many years ago was the same as you @zkarj, then a few years ago, I just decided to add anyone who wanted to connect and more recently I’ve partly reverted.
The reason for my reversion is down to the fact that when I read the timeline, it’s full of things and people I don’t care about, so from a position where I have approx 1,300 connections, in the last week I’ve reduced that to about half of that, and I continue to pick away.
Linkedin is an important way to network, learn, (at least in the UK) and a way to retain and strengthen ties with people. But it comes down to how you use it.
My final thought I had @Bmosbacker about your Tagline is that unfortunately from my experience, very few people who are speculating to try and sell you something read your Bio/description. They’re pulling your data and metadata as part of a list of potential contacts, they’re not looking at you as an individual usually. So they probably won’t see your tagline 
You are no doubt right, hence, my concluding statement, “
”. 
I figure the tagline will do a little good and no harm. My backup strategy is to ignore such requests (though I always feel rude doing so. If I decide to respond, I use my text snippet:
Thank you for your email and interest in serving . I have reviewed your material and have concluded that we have no need for your [product / service] at this time.
Although we do not have a need for your [product / service], we appreciate your interest in serving .
Please feel free to remove me from your [distribution / connection] list.
I use a similar text snippet for recruiters unless the message makes it obvious they didn’t pull a quality list or read my profile. I don’t want to insult or ignore people trying to offer me a job.
I did a career semi-change earlier this year and ran nearly the whole process out of LinkedIn messaging. Useful service.