Pictures will help a lot.
Getting perfect points is a skill level/work in progress thing. I used to be about being very precise both with my cutting and my sewing, and then about 10 or so years ago I started working with the idea of cutting/sewing large and trimming down for accuracy. There are things to learn about this way as well, like not all geometry is equal! But for me I getting better/more accurate results and I don't actually waste anywhere near the amount of fabric I was worried about but yes, there is some. I'm not talking huge amounts, just a 1/4" or so each piece and since I'm mostly doing strip piecing and using scraps it isn't an issue. Even with "planned" fabrics, I've never had to cut more than one extra row to make up for my cutting large.
When I can, I prefer to use the cut a square, draw a diagonal line and sew on the line. Trim excess and toss if it is small, make additional HST (half square triangle) units if it large and use for some other project or on the back. If you had heard the audible sniff of disdain frugal me would have done 20 years ago, you'd laugh at me now. For the time, the ease, the accuracy a bit of fabric is well worth the fun and better results I get now, even if I have to do a corrective trimming step each time.
Reality was I was checking my subunits for squareness and such things anyway -- now I just make it worth my time to handle the fabric and trim it instead of just shave a few hairs here and there.