Like Napoleon mentions in a sibling comment, you're typically not asking for a percent of profit directly, but rather for a weekly rate which is at a substantial premium to $50 an hour. (I've billed at $30k for a week before and pitched successfully at $50k.) A $60k engagement for my typical client wouldn't represent 1% of their profits -- many of them spend more on sandwiches in any given year. (If lunch as a perk costs $3k per employee per year and you have 40 employees... yep.)
Yes, it is extremely important that you're pitching the right sort of clients and that you're proposing to do something which meaningfully impacts their business. My typical client towards the end of my career was a B2B SaaS company with $10 to $50 million in annual sales. Bringing me on was generally not their #1 activity in any given year -- after all, they all have dozens of full-time employees, so they can do lots of things in any given year -- but it was generally for an initiative with a fair amount of strategic importance on, typically, a product which everybody knew would make serious bank if the initiative worked out.
If you had come up to me and said "We're building an app and need an extra Rails programmer" I'd have said "Cool, I keep a list of folks I like. Let me hook you up with someone." rather than attempting to get that business. Like everybody else who is Good With Computers, I occasionally get pitched on "Hey my aunt has a catering business, maybe you should make her a website?" That will never, ever happen. By comparison, if you'd recently experimented with A/B testing a bit on your $X million a year SaaS product and had increased sales by 5%, I would be very, very interested in helping you find the next 5 to 15%.
If you're continuing to find very budget-sensitive clients, how are you prospecting for them? How are you qualifying them? How are you pitching them?
There are many ways to prospect for clients which will get you lots of crappy leads, like e.g. looking for gigs on Craigslist.
Qualifying clients is an art. You don't have to proceed directly to the "prying" questions like explicitly asking what the budget for a project is. Just ask basic getting-to-know-you questions like what their core line of business is and how many employees they have. If their core line of business is retailing Beanie Babies and they have two part-time employees it is unlikely they can afford professional services. If, on the other hand, they have two dozen engineers on staff, they probably can write any check you can currently envision asking for.
What are you selling them on? If you're selling them things that can get delivered by any other freelancer who speaks your language, stop doing that. Start only going after engagements which, assuming project success, meaningfully increase the revenues of their company. Get case studies about how you've done that previously. Make it mandatory that you build in metrics tracking into all your projects, so that your clients know the ROI you are getting (and so you can quote, in broad terms depending on the specifics of your relationship and legal commitments, the sort of ROI your projects have recently generated.)