Someone on the Ars Technica forums realized what was wrong. I will bet quite a bit of money that you used The Unarchiver to unzip the file that contained the .mpkg file. Apparently, something about the way it unzips introduced some sort of corruption. If you go back, and instead of using The Unarchiver, use Apple's own Archive Utility, then try the installation using that newly unzipped .mpkg, it should work. It worked fine for me.
I guess I have almost the same situation as yours (recently reinstalled mac os, hence the newest version Catalina 10.15.4)
As @Mico and @egreg said, since you are "Mac OS 10.13, High Sierra, or higher", you should choose Downloading MacTeX 2020. After downloading, open Application/Tex you will see "MISSING APPS.pdf" where they explained due to the higher security requirement from Apple, there's some notorized issues. (More details here.)
Then, I just followed @egreg suggestion, download texlive from the developer's site
After installing texlive, it will say "“TeX Live Utility” can’t be opened because Apple cannot check it for malicious software." whenever you want to open it. The solution is: open with "right click with the pop up {open} choices".
Thank for you raising this questions. Hope that works for you!
Edit: Even if I successfully installed and even updated TexLive apps. The next time, I cannot find this apps in my application list. Instead, I need to search it with the upper right magnifier icon and opened the same way as mentioned above (right click).
Edit 2020/6/1 on Catalina 15.5.5: When I installed MacTex, it says "This package is fully notarized, as required by the Catalina version of macOS. Some GUI applications like TeX Live Utility and BibDesk are missing because they have not been updated for notarization. To obtain them, read MISSING APPS in /Applications/TeX."
Best Answer
It sounds really odd, but for some reason the MacTeX file can be corrupted through non-Safari browsers. Try downloading the file again but this time in the Safari browser.
I've run into this problem before, and just ran into it again now. However, this fix has now worked for me twice on two different computers and at two different times.