Writing a replacement is something I've pondered for several years, but the problem is you need the network effect. It's like starting a dating site (something I did once!) - if people visit and nobody else is there it becomes an empty graveyard of dead dreams.
Sure you don't need to scale to 6 million users immediately, but you need a good way to seed with users, and you need to make it captivating and interesting for those early users.
The other thing you have to consider is that LinkedIn is the master of dark-patterns. To contrast that you'd want to make almost as much as possible open. So export the resumes in JSONResume, or similar open fashion. Of course when you do that you remove a reason for your visitors to stay.
It's not an impossible task, but scaling and growing are hard.. I'm sure if you launched something it'd be popular, but would it be popular _enough_ to be worthwhile, and keep growing? Maybe if you launched on a per-country basis, or some other artificial constraint. (An alternative would be to allow signups to students and record course-histories, etc not just positions. That lets youngsters show their skills despite a lack of job-history.)
But when thinking about a new Linkedin, I feel like making a new social network with less spam, less clickbait, and a better UI isn't enough to make people switch. What you really need to do is make a strong community centered around a particular industry with serious news, occasional gossip, and heavily moderated comments. In other words for our industry it exists: it's Hacker News.
[1]https://www.linkedin.com/pulse/linkedin-deleted-my-donald-tr...
You are constantly spamed by career 'advices', recruiters, life coaches and people bragging about their work-related achievements... Its really difficult to find any interesting content there. I tend to spend no more than 10 seconds on it, unless I need to respond to a (recruiter) message. Feels like the only people getting anything out of Linkedin are those who are new to their respective fields, and need to network a bit and maybe land a new job. Otherwise, it feels like waste of Internet bandwidth.
Frankly I have such a low opinion about Linkedin, that I'm amazed that nobody came up with an alternative yet. Maybe its the whole 'professional social network' concept that feels weird to me? While I socialize with my colleagues on regular basis, I dont want _work_ to be at the center of my socialization. I prefer to talk to people about other things in their lives.