Relevanssi works just fine with shared accounts and doesn't need a VPS. It's the size of the database that's the problem. Relevanssi is built for smaller databases. Larger databases are heavier to handle, and thus need more robust, professional-level search solutions.
Those solutions, unfortunately, tend not to be free. Why should they be? Developing and hosting them certainly isn't free. Nothing idiotic about that.
Relevanssi is free, but I'll be the first to admit it doesn't do everything, and I would not recommend using Relevanssi on large, high-traffic sites, as it is simply not built for such specs. Like I said - 14 000 posts can be too much for Relevanssi, and you've seen that first-hand. So, you'll have to look for another search solution. Relevanssi is not the only WP plugin available to enhance search, so maybe another plugin will work better for you.
While your aim may be to get everything for free, I'm myself more than happy to pay for premium themes and plugins that work well and do what I need. I find it absurd for someone to want professional work and expect to pay nothing for it.