What happens if you run this manually?
/usr/local/bin/curl -I
https://acme-v01.api.letsencrypt.org/directory+ FULL_NONCE=
I actually had a similar issue recently where I got a libcurl error when running curl - it wasn't shown because of the silent flags. In case you *do* also have the libcurl error when running curl you probably have the same issue I had - that is that when running a 32-bit architecture the library path is loaded in the wrong order.
The package libc-bin will add the config files in /etc/ld.so.conf.d/ and load them in alphabetical order. the file libc.conf in that directory will load /usr/local/lib and the file <architecture>-linux-gnu.conf will load the other library paths. in case of x86-64 it will be loaded after libc.conf - but if you have i386 / i486 it means libc.conf will be loaded last so curl will use the wrong libraries.
I fixed this by doing the following:
cd /etc/ld.so.conf.d/
ln -s libc.conf 00-libc.conf
ldconfig
After that it worked correctly again for us. It's actually a bug from what I could find that never got fixed:
https://bugs.debian.org/cgi-bin/bugreport.cgi?bug=685706
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Your issue may be entirely different, but I had the exact same recently and this was the issue for me.
Let me know, interesting to see/hear if others encounter this since I did not expect it
I haven't had those myself - if you are more specific with the actual errors you are getting and when they occur perhaps someone here can help out