The first indexable tool you showed is using a CCMT 32.5 style insert, I use the same tool quite often with CCGT inserts. Not sure if you actually did try using that for threading but it's only good for facing & turning since it's an 80° diamond shape insert, not good for 60° thread forms.
If you are threading soft aluminum or finer threads on steel, instead of the modified flank method (feeding with compound set at 29-29.5°), try threading with just feeding the cross slide straight in. The modified flank method is the better way to thread (arguably) since it's less wear on the cutter, less tool pressure, etc., but if you're not threading a coarse thread you can get away with feeding straight in with the cross slide on a lathe this size. Align the tool bit tip with the work using a fishtail & give it a try.
You mentioned jagged/sharp threads. If you were using the modified flank method to cut threads, that's usually a good indication of the compound angle set incorrectly. This is a common mistake with Asian lathe owners new to threading. The compound must be set 29-29.5° off the cross slide or 61-60.5° off the spindle axis.
Many (but not all) Asian import lathes do not come with a full protractor for the compound slide like older American lathes do. Most of them indicate 0° when parallel to the spindle (like the PM1236 does, I have the same lathe). So if you set 29° on the scale you are at the wrong angle for threading. You will need a protractor to set the correct angle for threading as the scale on the compound slide is useless for that.
This is what 29.5° looks like using the scale, incorrect for external threading.
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This is what 29.5° looks like for threading.
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As you can see the scale is useless for that on the PM1236.
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I use a preset slide bevel to set the compound for threading.
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