I am an employer; I own and operate a software company. I completely understand the sentiment behind your company's statement (though I would express it differently than you have summarized it here).
One thing companies see from this side of the negotiation is that a surprisingly large number of people see companies as a commodity resource. This is summarized by statements like this, "I can earn $75K here or $80K at company X. I guess I'm leaving!" In this case, the company is being viewed as having nothing unique or valuable in itself; it is a means for trading some labor for some money.
Depending on the company, this can be profoundly off-putting. I realize that when I get a new employee, they didn't suffer through the years of skipped paychecks, foreclosure notices, 120-hour work weeks (yes, really), hardship on my wife, and the years of my childrens' childhoods spent wanting rather than having. I get that, for sure. But all of that was done in service to a dream. At some point in the past, your employers have poured their hearts and souls into building their companies, and risked marriages, health, and (much more importantly) years or decades of their lives to try to build something-- generally to be demonized for it ("greedy owners", "the 1%", etc).
The statement you gave can alternately be translated as something like, "We really, really want to make this a fantastic place to work for people, and we don't mind paying you well... but we have to take precautions against the employees who are not willing to labor alongside us; we can't always tell which is which until a few years of service (usually around 4-6).
On the other hand, my goal is to try to pay new employees a base starting salary that puts them in the 85th percentile of their job-peers. If I find someone good who's really into it, the way I have been, and after we establish trust that we're both all-in on this team, I'll generally pay them whatever they want, even if it's a 10%, 25%, or 100% raise. If they are great, they are worth it (not just to the company, but to me personally--who wants to waste his life working with terrible people?).
All that said, here's my advice: Go in and treat them like the people they are. Tell your boss or whoever, "Hey man, I really like working here. You all have made this a great work environment with a product/service I feel like I can really be proud to be a part of and I plan to be here a long time. You've seen me take time off, and you've seen me work 80 hour weeks for a deadline. You know I'm all-in on this and I just wanted to tell you that I really love it here. In my personal life, I'm working on a few dreams that my family and I have and one of the keys to getting that is for me to be able to work at $xxxK/year, which is a goal of mine. I would love to be at that pay rate now, or even reach it over the next 3 years. Does that seem do-able?" Don't be arbitrary about that number; don't base it on the market rate for your job. Base it on your real, live dreams. Want to travel? Want to take six months to work remotely from Nepal? Want to buy a boat? Want to take your family on a mission trip to Sudan? Base it on those goals, and how you can achieve them. You don't have to tell what the dreams are, but be a real, live person with a real heart and soul -- and talk to your boss like he's a real, live person with a real heart and soul.
If they say yes, then great. If they say no, then ask them how that could become a reality. Take on new responsibilities? Done. Mentor some of the newer employees? Done. Do I need to transfer to a different role (that still fits)? Done. If none of that works... well, you've gotten clarity about your dreams and you should go find a company that's a good fit =) Not a commodity paycheck mill, mind you. A company that's a good fit with your life and dreams.
I hope that helps.