Good freeware NLE editor?

So sorry for the late reply, months, in fact. But when I was dealing with this issue, it was at a time when I was dealing with bereavement and I just couldn't handle another complicated issue.

Sorry about those circumstances. Bereavement. Learned a new word today, both beautiful and sad... not quite as chilling as this one although it can come close...
And I know the feeling about having sudden, unexpected, prolonged leaves of absence.

Can you recommend how to remux, then, please? I've read a bit about it today, but it seems complicated. Hoping that with the right software, it'll be easy. Any recommendations, by any chance?

Depends which formats are available for export with either of those NLEs (Grammar Nazi aside: “NLE” stands for “non-linear editor” so “NLE editor” is redundant). If it's MPEG2 only, that's a very inefficient format by today's standards (as in, it can yield good quality but with a much higher bitrate than more recent formats). Also I would be surprised if MP4 or AVI containers can handle MPEG2 video with no hiccup. For sure it's not standard. (MP4 nearly always contains AVC [H.264] or HEVC [H.265] video and AAC audio streams; AVI nearly always contains either uncompressed video which is huuuge, or lossless compression formats like Huffyuv / Lagarith / UT Video / MagicYUV which yield marginally less huuuge files, or lossy compression formats like Divx / Xvid which are all pretty much obsolete at this point, and AC3 or MP3 audio streams which are pretty much obsolete too – and the AVI container itself is considered quite antiquated by now, mostly used for intermediate files, not for “delivery” files, only fools still encode movies in Xvid + MP3 in AVI container.)
MKV on the other hand can handle pretty much anything, but it depends what you intend to do then with those videos.
Remuxing can be done quite easily with a simple “set-and-forget” ffmpeg script, like for instance, on a Windows system:

Code:

FOR %%A in (*.mpg) DO ffmpeg -i "%%A" -c copy "%%~nA.mkv"

saved as a .bat file, then a double-click on that .bat file will losslessly remux each .mpg file found in the same directory to a MKV file with the same name (for it to work the ffmpeg.exe executable must be either in the same directory, or in a directory registered in the “PATH” system variable; or else the full path to the location of the ffmpeg.exe file must be specified in the command).
But again, it might not be an ideal solution depending on the purpose, especially if you're going to do video editing more regularly, it would probably be better to get acquainted with something more up-to-date. (Still, learning the basic of ffmpeg and remuxing could certainly come in handy down the line – as for why that line always has to go down, your guess is as good as mine.)
If either of those allows to export in a lossless AVI format (like those mentioned above – each one has to be installed first for it to be listed among the options, years ago I've been recommended either UT Video or MagicYUV rather than HuffYUV as they're more efficient while still easy on system resources, I've used Lagarith which in my experience yielded the best compression ratio out of those four but is reportedly more likely to cause performance issues when editing, don't know what's considered “best” by now), then the output can be compressed with a single lossy compression step, and it can also be done with a “set-and-forget” ffmpeg script, for instance:

Code:

FOR %%A in (*.avi) DO ffmpeg -i "%%A" -c:v libx264 -crf 20 -preset veryslow -c:a aac "%%~nA.mkv"

likewise, saved as a .bat file, then double-clicking on that file will convert each .avi file in the same directory into a MKV file with an H.264 video stream and an AAC audio stream (specific settings can be tweaked to reach an optimal size / quality ratio; in this example I used “-crf 20” which is generally considered a good compromise in that regard; for audio I didn't add specific settings, not sure what the default settings are by now, for my own use I encode the audio separately with qaac as it's reportedly better than the AAC implementations available in ffmpeg, at least that was the consensus some years ago when I did some research on that topic). But that solution requires a lot of temporary storage space (an hour of losslessly compressed video can easily be over 100GB, and I've mostly done editing in 1280x720 resolution).

As for the original question, there must be good options by now when it comes to free NLEs, but I have no first hand experience, I'd be interested too. I've briefly tried DaVinci Resolve, which if I'm not mistaken comes as a fully functional basic free version, but I found it quite difficult to handle (that thing is used in professional workflows). I've read about Shotcut and Openshot but haven't tried them. Hope someone will chime in.

I would give Shotcut a try. Checkout their website and take a look at the tutorials to get a first impression. Pretty sure it meets all your needs for an NLE.
https://shotcut.org/

Kdenlive is similar to Shotcut. I find it a bit less intuitive perhaps.

For simple remuxing an ffmpeg commandline is easiest though - unless you are frightended by commandlines. There are many ffmpeg frontend GUIs though, like ProWo's clever FFMpeg-GUI for example
https://www.videohelp.com/software/clever-FFmpeg-GUI

Also, AVIDemux might be an option for you, although its development seems to have come to an end (?)
https://www.videohelp.com/software/AviDemux