Greetings.
I have several machines, and have thread cutter, needle up/down, needle threader on some. I love them all and use them all. But I too have had some issues with the thread getting into a nest when starting the next seam after cutting. This is especially common when starting on a point (triangles, not squares). I know that people use leaders/enders for this reason, but I tend to forget.
One thing I saw at a quilt show vendor booth and bought on impulse, and it turned out to be a lot more helpful than I expected, was a little thread cutter "button" that you attached to your sewing machine. After finishing a seam, pull the sewn piece out from under the presser foot, and up and over the button to cut the threads. That way, you can put the cut where you want (as close to the end of the seam as desired), and you'll be holding the next set of thread tails to keep them from getting sucked into the machine when starting the next seam. You can use the same "cutter button" on the longer tails after you have sewn enough of the next seam for the long threads to reach the cutter button. Voila, all threads neatly trimmed as you go, no need to go trimming after the quilt is pieced.
I attached the adhesive-backed thread cutter button to the side of my machine, about half-way up the side. This gives me two choices for how to cut the threads--automatically via machine, or manually via add-on button. I liked the arrangement so well I bought a couple more of these thread cutter buttons for all my machines.
Here's a link to a place that sells them. (I did not get mine from them, I have not done business with them, but it was one of the few places online I found that had a good picture. No affiliation, yada yada.)
https://www.jukeboxquilts.com/shop/N...-x24853865.htm
...dany
PS--Re your original question, I would definitely recommend as many of the bells and whistles (needle up/down, needle threader, and auto cutter, in that order) as you can find in your price range. Good luck in the hunt.