Core 2 Duo is still viable for everyday use, ie web browsing, watching videos, typing up Word documents. My mom and my brother both still use Core 2 PCs. Both the same model, Dell Dimension 9200s that my dad was able to bring home from his office a few years ago back in 2011 when the company upgraded to new computers. They were originally specced with Core 2 Duo E6600s 2.4 GHz, 4 GB of 667 MHz DDR2 RAM, Radeon X1300 Pros (yeesh), and standard SATA II HDDs. We've since upgraded the PC my brother uses to a 2.4 GHz Q6600, Radeon R9-270X, 8 GB of 800 MHz DDR2, and a Samsung Evo 850 SSD for the boot drive, and replaced the PSU. Basically the only components original to that PC at this point are the mobo, the case, and the DVD drive. Upgrades to the PC my mom uses have been less extensive, really just replacing the PSU and slotting in one of my older GPUs I had hanging around, a Radeon HD 5770. Both run Windows 10.
One thing that annoys my mom about her PC is that video playback on Facebook tends to repeatedly stutter and hitch (this is with Chrome). Playing video directly from YouTube is fine, though -- the issue only occurs when the video is embedded on other websites. So I'm not sure if that's an issue with the CPU speed just not cutting it, or if it's a software/driver support issue for the browser's hardware acceleration with the 5770 on Windows 10. I haven't extensively tested the issue, and I haven't bothered trying recent games on it.
I have done more extensive testing with my brother's PC. It is viable for gaming -- to an extent. My brother uses an older 1440x900 monitor, so that's what I've tested it at. The stock Q6600 certainly bottlenecks the 270X, but Mantle/Vulkan/DirectX 12 breathes some new life into it. Dragon Age Inquisition, Doom 2016, and Deus Ex Mankind Divided all get a boost when using the low-level APIs, able to be run at mostly high settings. DAI sticks close to 30 FPS, matching console performance (though DAI's Mantle renderer has frametime issues on Windows 10 that were never fixed, which can be mitigated by enforcing a 30 FPS cap with command line arguments). Doom reaches between 50-60 FPS in most situations, though one level, the Argent Tower, seems to hit the CPU especially hard, requiring shadows, a CPU intensive setting, to be lowered from medium to the lowest setting. Upon doing that, though, it's smooth sailing once again. Deus Ex Mankind Divided fluctuates between 20 and 30 FPS in the larger, NPC heavy city areas. Interestingly, lowering shadow quality doesn't seem to improve DXMD's framerate like it does with Doom. Steam's frame rate counter and MSI Afterburner's frame rate counter actually indicate that DX11 provides higher framerates than DX12, but my subjective impression from playtesting is that DX12 is much smoother and less hitchy than DX11. I have't been able to objectively test that though, as Deus Ex lacks a built-in frame time monitor like what DAI and Doom have. (worth noting is that external software frame rate counters may be inaccurate -- Dragon Age Inquisition and Doom's built in monitors and their results tend to differ from external counters). In any case, GPU usage goes up from 40% to 80% between DX11 and DX12 in DXMD, indicating that DX12 mode is much less CPU bottlenecked on the Q6600.
So that's the testing I've done. My brother also plays a lot of NBA 2K16 on the PC, and is able to play at smooth framerates on high settings. Point is, a stock Kentsfield (quad core version of Conroe) is able to approach the performance of current-gen consoles, especially when using low-level APIs. Its successor, Yorkfield, should be even better. And the answer to the question "do you junk or pass on the Core 2 system to another person" for me is yes, at least a couple years ago, as I basically passed on one of the Dimension 9200s to my brother once I had put the Q6600 in it.
When it comes to Core 2 systems, the real issue may be the longevity of the motherboards rather than the performance of the CPUs, like others have said. I hadn't even thought about that before. Neither PCs' motherboards have gone out on me yet *knocks on wood.*