I don’t have any solutions. Sadly, I can only corroborate your experience. Bear in mind that Shotcut is built on top of ffmpeg. Decoding issues like you’re experiencing are probably from ffmpeg, not Shotcut. I have had very sketchy experiences working with .TS and .MTS files even directly with ffmpeg, so I would expect those problems to ripple up to Shotcut. It’s so bad that I have an ingest process that converts .TS and .MTS to an intermediate format with the -async 1 -vsync cfr ffmpeg options to eliminate the stutter. Transcoding like you’re doing with HandBrake is the only fool-proof solution I’ve found so far. There are lossless formats like FFV1, HuffYUV, and Ut Video if you’re concerned about preserving absolute video quality. Those transcodes won’t cost you anything except time.
The alternative, of course, would be capturing in a format other than .TS if possible.