Sounds to me like the tool is functioning as I would expect it to. Is it telling your PC isn't "eligible" for, or isn't "ready" for or "compatible with", Win10? The former would imply (to me) that you aren't eligible to "obtain" the upgrade installation program/iso again, because you've already done so... Not "ready" for, or wording along those lines, would suggest that MS suddenly changed Win10's system requirements, which seems highly improbable, to say the least.
As for burning a DVD, well, you should certainly back up the installation file you previously downloaded, but it doesn't much matter what medium you choose for that... DVD, flash drive, either better than cloud storage in case your PC goes bottoms up before you get around to installing the upgrade. But I assume you can also get some sort of back-up copy/restore disc for Win10 itself with your key, though presumably not via the same method you used to "obtain the upgrade"...