Most people I know would simply say 9:45 instead of 1/4 to 10 when looking at an analog clock. We were taught to mentally do some minor math, same habits, just better I guess.
Just different and from a different school of thought. If you consider the convenience of conveying how close the time is to the hour, it’s actually a quick shorthand. I can see you don’t like it however it’s just the way people think of time. It’s not like they can’t do the math because if they have a digital clock they have to convert 9:45 into a quarter to 10. It’s the way they think about time.
View it from their perspective, they all converse perfectly together because that’s how everyone told the time however other people want to change the rules. They aren’t being rude or impolite just using the system they grew up with.
For me, it's because when I look at an analog clock, it's easier to see that there's a quarter of the cycle left on the minute hand than to convert 9 to 45 mins. So if someone asks me for the time and I looked at an analog clock, I would be more likely to respond with the relative position of the minute hand than with the exact minutes. If I am looking at a digital clock, then I will read exactly what the clock says.
This is the correct explanation. Having grown up in a house full of analog clocks and preferring them still today, it's hard for me to understand the difficulty so many people have using analog, or why it's hard to interpret the logic used in conveying it. But it really is just a matter of immersion and perspective. People who prefer analog appear to perceive time in fractions and circular cycles. Half to/past, quarter to/past, etc. are how we actually view and perceive the day. It's more than just a number.