It's more about being "independent of advertisers" than whether advertisements are practically hosted on the site. An archive should be a third-party, neutral source, and advertising jeopardises this.
It's not a newspaper (as only one example) where they hold editorial control over the content and are therefore (in theory) beholden to the people who pay the bills (the advertisers). Forgetting for a second whether it would have any major impact (I say it would not I mean they scrape web pages in a very clear fashion) it's hard to believe it could go the route of say YELP in their mission. Or that a minor impact to what they do would not be offset by not having to beg for money. (With PBS it was called a begathon when they had to raise money).
And even with donations they could in theory be 'corrupted' just the same. A rich person could give them a large sum of money (as a donation) and then would have some defacto say in how things were done.
Take as another example ball parks (to even counter my point). They sell naming rights. That does not mean that the entity who purchased the naming rights gets to decide who plays on the team or who coaches the team although you could argue that that could happen (and I would say it does not offset the benefit of not having that 'independence'.