IANAL, but I believe the following is all true even if SCOTUS finds an argument to overturn the CO Supreme Court ruling (which, if they do, I believe will not be on these grounds).
[1] A conviction for insurrection is probably (almost certainly) not constitutionally necessary. While a conviction would certainly make the argument for denial of ballot access easier, the 14th Amendment does not textually require a conviction, nor is there strong evidence that an originalist interpretation would lead to the conclusion that a criminal conviction is required.
The Amendment bars anyone who "engaged in" insurrection or rebellion and makes no stipulations as to how their culpability must be determined.
[2] Trump was not denied due process, nor was the determination that he engaged in insurrection made with no standards or on an "everyone knows it" basis. The Colorado lower court conducted a 5-day civil trial or trial-like hearing at which Trump's lawyers could present evidence and argument, and the judge used a "clear and convincing evidence" standard entirely appropriate to a civil case.
No criminal penalties attached to the judge's ruling that Trump engaged in insurrection, so neither a jury trial nor a "beyond reasonable doubt" standard were constitutionally required.
The Colorado Supreme Court did not determine Trump's culpability on their own, but treated it as a finding of fact by the lower court. (Appellate courts normally accept findings of fact as given and don't engage in fact-finding themselves - if they have reason to doubt factual findings, they'll normally send the case back to the lower court with orders to engage in further fact finding.)