Can I shrink a GIF by converting it to MP4?

That is exactly what the GIF to MP4 tool is for. Find it in the Tools menu, upload a GIF of up to 50 MB and convert it - our servers encode an MP4 that is usually far smaller than the GIF you started with and plays on pretty much any device. Download it when it is done. Like everything else here, it is free with no sign-up.