My guess is that it stems from the old saying “you get what you pay for”, and as such a free service without a megacorp behind it MUST be dodgy and untrustworthy.

That's my guess too. I think a lot of non-techie ppl have been burned by "freemium" apps that are steaming piles of shitware. It's kind of a hard distinction to explain, between "free" as in reputable OSS, and "free" as in spyware and thinly disgused malware. So they go wtih brands with name recognition. Like, you know, Facebook. Or Google.