They only want to pay for time spent, and that's it.
No, they're being cheap and unrealistic. I have plenty of clients (including both business and residential) who would not think, for a single second, of not paying me if they engaged my services to fix a problem and it only took 3 minutes to fix.
Skilled trades and professions are not hourly workers in the sense of pay exactly the time it takes. There are always minimums "per service call."
Essentially, I'm not letting the outrageously stupid/cheap/nasty client off the hook even as easily as you are. I don't expect that anyone I engage to perform a service is likely to charge me for 5 minutes time even if, in a given instance, that's what it took. There is a service call minimum (and although even I'll admit there have been times I've deeply resented having to pay that, I kept that to myself).
One of the reasons (among many) that lawyers are so hated is that they do not give you a single second of their time without billing for it. I don't operate by that business model. If I have a loyal customer send me a text or email that I can answer virtually instantly, that's a freebie. The amount of additional referrals and good will I've generated by taking this approach far outstrips what I would have gotten in billed time.
I take the middle road, but as I said earlier, I'm the one who's making the call about when a freebie is doled out, not the client. I've actually had several people pay me recently just for having answered a question (and where I did not bill/invoice and was not going to). There is great business value in strategically extending helping hands without expecting payment, but you have to decide when those are occurring.