I can definetly do this. I have a lot of experience with scraping YouTube. I even have specific experience with YouTube Transcripts(converting them to SRT).
Now the part about "download parts of the video" is a problem. The site in question will only allow you to download the entire file, not parts of it.
Searching through the YouTube transcripts, and downloading the videos, can surely be done within a 100$ budget.
But cutting up the videos, and joining them into one, is something i don't think will be that easy to do (unless you are on a *NIX system).
User interface <- I can do a web, CLI(or ncurses) interface. A fully blown GUI is something i've never tried in Python, but i'd be willing to learn. I suspect it is pretty straightforward. I just don't personally use GUI's that often.