If you were aware of sample size determination, why would you suggest that a 6000 person sample is "pretty small"? It's pretty big for a national study, actually.
Whether or not the sample selection was properly done with attention paid to bias is difficult to say, but has little to do with the sample size since a non-randomly-selected sample is likely worthless whether it's 6000 or 60,000.
I have some significant concerns about this study, but sample size is not among them.
Edit: also, I disagree that sample size is entirely worthless in non-random sample selection. If your sample size is a big enough percentage of the target population you are studying, you can draw some conclusions regardless of randomness. You just end up having to use absurdly large samples to get meaningful data.
The issue is randomness. If it's a random sample, it's very large. If it's not a random sample, it's worthless.
"We have no idea how they found the participants or controlled for bias in the selection process."
A) This has nothing to do with sample size
B) If they have any credibility whatsoever, they hopefully know what they are doing. Adjusting is a statistically legit thing to do, they do it in political polling (and in hindsight) it works.
But yes - with 'online surveys', it's hard.
Once you're over 60 - 300 survey participants, you're well within large-number stats territory. If you're trying to estimate multiple points of freedom, you might have to increase your sample size.
Simply drawing a larger sample of a biased sampling methodology isn't going to solve any problems. You'd have to aggressively correct for errors.
The survey here was conducted by a firm dedicated to such work (a positive sign), but it's attached to a large public-relations firm (Edelman), which I consider a very strong negative. It isn't a disinterested research organisation or dedicated polling firm such as Pew or Harris.
Edelman's other clients include Microsoft, Johnson & Johnson, Pfizer, HP, and Samsung. Quite the list of rusted stars.
In the research, more than 6,000 U.S. working adults over the age of 18 were surveyed online by independent research firm Edelman Intelligence between July 29 — August 24, 2016. Among them, 2,049 were freelancers and 3,953 were non-freelancers. The survey defined freelancers as individuals who have engaged in supplemental, temporary, project- or contract-based work within the past 12 months.