Why is there a package available at your site that one does not see if one looks for it?

Which above page? You have linked in many pages to your question...

see the publishing history https://launchpad.net/ubuntu/+source/nvidia-graphics-drivers-304/+publishinghistory

nvidia-304 version 304.131-0ubuntu5 was superseded by
nvidia-304 version 304.132-0ubuntu1
This explains why it will no longer be listed on https://launchpad.net/ubuntu/+source/nvidia-graphics-drivers-304

If version 304.132-0ubuntu1 does not work, then please create a bug report with the details.

@actionparsnip
The file: https://launchpad.net/ubuntu/+source/nvidia-graphics-drivers-304/304.131-0ubuntu5/+build/10627168/+files/nvidia-304_304.131-0ubuntu5_amd64.deb
is not visible at the location: https://launchpad.net/ubuntu/+source/nvidia-graphics-drivers-304
even though it is there, downloadable, and usable.
I only learned of ...ubuntu4 and ...ubuntu5 accidentally from a changelog of nvidia-304, and only learned the the file was available at the site by reading a comment entered for a bug I was researching.

@Manfred Hampl
It may have been superceded, but the even earlier ...ubuntu3 is easily seen there as "supported" under "Xenial Xerus"

The version 304.132-0ubuntu1 will not load into the kernel. A bug report:
https://bugs.launchpad.net/ubuntu/+source/nvidia-graphics-drivers-304/+bug/1634802
has already been filed (I have commented there, and also have a duplicate bug on that).
They have made available an updated version: 304.132-0ubuntu2 .
This loads into the kernel and seems to function (for me - some others still have problems, it seems)
BUT the panorama program Hugin, which starts up fine under Nouveau (but under which Nouveau itself consistently crashes)
crashes with some sort of X error.
The superceded version 304.131-0ubuntu5 is the ONLY version I have tried that both loads into the kernel, AND with which Hugin loads and runs without problem.

The ONLY reason I am not dead in the water when it comes to using Hugin, is through the two two accidents mentioned above..
Otherwise, I would never have known that 304.131-0ubuntu4 and ...5 even existed, let alone known that the latter was available at " https://launchpad.net/ubuntu/+source/nvidia-graphics-drivers-304", but hidden...

I guess my main point is, that older versions should be clearly visible and available for download, so that if the "new and improved" version doesn't work for someone, he can at least try an older one that might work for him.

The version that works for me is the superceded version, which is not visible at the main page,
and I suspect that it cannot be reached by clicking links from that page.
(I have no idea how the person who showed the link knew it was there and how to reach it
See comment #9 at "https://bugs.launchpad.net/ubuntu/+source/nvidia-graphics-drivers-361/+bug/1573508")

For each Ubuntu release there is a maximum of five versions of a package listed on the overview page:

The version that was published in the original version of the Ubuntu release ("release" in the second column).
A version that was created to help for security-relevant bugs ("security")
The latest updated version provided for this Ubuntu release ("updates", only in rare cases different from the "security" one)
A version in the verification and test status ("proposed")
And a version with an updated version that was backported from a newer to an older Ubuntu release ("backports")

(Usually not all versions will be existing for a certain package, the only one being mandatory is "release".)

If there is already an "updates" version, and a new updated version is published, then the older version will not be shown on the overview any more.

But as already written:
All versions (including the superseded ones) can be reached via the link "View full publishing history" in the top right corner of the overview page for the source package, in this case: https://launchpad.net/ubuntu/+source/nvidia-graphics-drivers-304/+publishinghistory

Have you tried switching to a higher version of the NVidia packages, e.g. nvidia-graphics-drivers-340 or nvidia-graphics-drivers-352?

The on-motherboard graphics is reported as
lspci -vv ==>
00:0d.0 VGA compatible controller : NVIDIA Corporation C61 [GeForce 7025 / nForce 630a] (rev a2) (prog-if 00 [VGA controller])
 Subsystem: ASUSTeK Computer Inc. C61 [GeForce 7025 / nForce 630a]
 Control: I/O+ Mem+ BusMaster+ SpecCycle- MemWINV- VGASnoop- ParErr- Stepping- SERR- FastB2B- DisINTx-
 Status: Cap+ 66MHz+ UDF- FastB2B+ ParErr- DEVSEL=fast >TAbort- <TAbort- <MAbort- >SERR- <PERR- INTx-
 Latency: 0
 Interrupt: pin A routed to IRQ 21
 Region 0: Memory at de000000 (32-bit, non-prefetchable) [size=16M]
 Region 1: Memory at c0000000 (64-bit, prefetchable) [size=256M]
 Region 3: Memory at dd000000 (64-bit, non-prefetchable) [size=16M]
 [virtual] Expansion ROM at 000c0000 [disabled] [size=128K]
 Capabilities: [48] Power Management version 2
  Flags: PMEClk- DSI- D1- D2- AuxCurrent=0mA PME(D0-,D1-,D2-,D3hot-,D3cold-)
  Status: D0 NoSoftRst- PME-Enable- DSel=0 DScale=0 PME-
 Capabilities: [50] MSI: Enable- Count=1/1 Maskable- 64bit+
  Address: 0000000000000000 Data: 0000
 Kernel driver in use: nvidia
 Kernel modules: nvidiafb, nouveau, nvidia_304
The nvidia website, and "Additional Drivers" both mention only the nvidia-304 for this.

nvidia:
" The 304.xx driver supports the following set of GPUs:
NVIDIA GPU product Device PCI ID
GeForce 7025 / nForce 630a 0x03D6"

I was assuming that the higher-numbered packages were only for more advanced graphics chips/cards.

[I did actually try a higher numbered driver, but when I did, it failed to load into the kernel just like the original 304.132-0ubuntu1.]

You are right. The Readme of the 340.98 driver tells that the GeForce 7025 / nForce 630a 0x03D6 is supported only by the 304.XX driver. I was not aware of that.