I'm not a lawyer, and I was not involved in drafting any GPL license, but below is my common-sense reading of the licenses, starting from an assumption they are legally sound.
Shouldn't section 1 of LGPL v3.0 provide exception to section 5 of GPL v3.0 instead?
Short answer: No, because section 5 contains a permission, not a requirement. The permission has requirements, but to eliminate section 5 would be to eliminate the permission altogether. By contrast, section 3 is entirely a requirement, which as a special exception does not apply while you exercise your permissions to combine and share the library under LGPLv3 sections 3 and 4.
The GPL contains a set of permissions. The LGPLv3 is a set of additional permissions on top of those present in the GPLv3. You assume that the more lax LGPL would need to limit the restrictions present within the permission to convey modified versions in GPLv3 section 5, but this is not the case. Instead, the LGPLv3 simply adds a new permission to share modified versions that stands alongside the more limited permission in the GPLv3.
That is, an LGPLv3-licensed work includes permissions such as
If you modify a copy of the Library, and, in your modifications, a facility refers to a function or data to be supplied by an Application that uses the facility (other than as an argument passed when the facility is invoked), then you may convey a copy of the modified version:
a) under this License...
And it also includes (i.e., inherits from the GPLv3) the completely separate permission
You may convey a work based on the Program, or the modifications to produce it from the Program, in the form of source code under the terms of section 4, provided that you also meet all of these conditions:
c) You must license the entire work, as a whole, under this License to anyone who comes into possession of a copy...
If you want to distribute the work, you choose which permission you'd like to exercise: you are free to choose to exercise a more limited GPL-supplied permission, or you may instead do so under a more lax LGPL-supplied permission, if your distribution meets the LGPL's preconditions.
To highlight another issue with your interpretation (as far as I understand it), LGPLv3 section 1 only applies to sections 3 and 4 of the LGPL. It does not apply to section 2, which would also need to have the requirements of the GPLv3 held back, if your interpretation were correct that GPLv3 requirements always apply to redistribution unless the LGPL explicitly exempts them.
Instead, LGPLv3 section 1 really does seem to eliminate the requirements of section 3. That is, downstream redistributors who share the library in the ways allowed by LGPL sections 3 and 4 may assert that their work is subject to effective technological measures -- i.e., if your software uses an LGPL licensed library, you may still take legal action against users who circumvent DRM in your software, if such circumvention violates laws against DRM-circumvention. Without this permission, section 3 would bar you from taking such legal action.