You raise an interesting question, as well as causing me to discover the latest updates from our friends at Adobe.com.
First, what computer type are you using? If a 64-bit Mac, you're already out of luck for the old LR6. Given you're still running this old software, like I am, I'll guess you're on Windows.
What I just discovered is that Adobe has now disabled the LR6 download from 'my products' at Adobe.com. I can see my purchase history of LR5 and LR6 (upgrade). But now, the software is no longer available for download from these 'official' locations.
When I check my System Info from inside LR6.14, my software reports both a system and application architecture as x64 (64-bit). I also have an old backdoor to the LR6 (Win) software at Adobe that still works. Consider
immediately downloading and saving this link as you decide what to do next (to use this link, you have to have an active sign-on session at Adobe.com):
Windows x64
http://prdl-download.adobe.com/Lightroom/60BA1251F1BC48B8B82B1B63AE8E620E/1552643270580/Lightroom_6_LS11.exeRegarding LRCC, do you mean some legacy 2017 software? Or, do you mean an active LR/PS subscription? In today's Jan 2022 subscription world, the LR 'classic' software comes with the subscription, software that downloads and installs to your local harddrive, operates on your local computer and uses images located primarily on your local storage. The same applies to PhotoShop.
The subscription also gives access to LRCC software that operates on your mobile device(s) in a browser, something that you can also use in a browser on your main desktop. In the basic subscription, you also have access to 20GB of cloud storage that would let you wirelessly share images between your mobile device and your main desktop.
The browser LRCC has a lot of power, but not all the power of the classic software. Adobe seems to have created a LRCC only subscription option, but that's really for the all-mobile, I don't have a wallet, my phone is my computer Gen-Z crowd. Based on what you've said above, you're probably not the target demo / use-case for this option.
If you have a 64-bit computer that doesn't have LR6, consider testing the download provided above. Don't screw around with your active LR6 software until you've done some verifications of the software link provided.