I try to lead by good example and say instead that "currently, I can't see myself taking the time to do" something. It's a small change but it feels a lot more honest.
The "have time" phrase is an idiom[1]: a group of words established by usage as having a meaning not deducible from those of the individual words (e.g., rain cats and dogs, see the light ).
For the criticism you wrote about "have", someone else can write the same criticism about "take" in "taking the time". And the same defense can be used for that too: "take the time" is another idiom that must be interpreted as a group and shouldn't be logically parsed as separate words.
If I don’t have time for something, I don’t have the remaining hours to spend on it, given the time I’ll spend doing other things.
So no, you never just have time you're not using anyway. You're always using all of your time. On what is a matter of priorities.
However, if you would spend 19--21 watching a TV series, or twiddling your thumbs, you can choose to take that time to do something else.
(Another one of these is that I try to never use the word "empower" but explaining that one is a bit more complicated.)
When I say “they consider”, I don't propose that I know what's in people's minds. My experience is that people don't talk about their time as something they choose to allocate in a certain way. Rather, they talk about it as if it's fixed, like laws of physics.
You can very well say “I can’t afford that” or “I don’t have money enough to buy that” even if you technically have money in the bank enough to pay for something right now. Maybe you want some money as a buffer for unplanned expenses, or you’ve already agreed to but something that’s not yet paid.
Same goes for time: Yes you can choose what to do at any given moment, but the decision may already be made. You may have promised to do something - meet someone for lunch, go to work. Or you know that you need that time to sleep in order to feel well.
Surely it’s possible to change those decisions, the same way it’s possible to spend money in a restaurant instead of paying the rent. It’s not weird to speak of the decisions as final and the time already “spent”, though.