First, all throughout Fate, ideologies clash, and one is supported over the other. But it's just in the moment, and usually is to poke holes in the one being shat on, and not to suggest that the other way is better. It all happens pretty cyclically, where all the different parts of everybody's ideologies get shat on in one place or another. Fate, when taken globally, hardly ever espouses a viewpoint as being actually 'correct.' With that context, it's supposed to be almost taken for granted that the Banquet of Kings isn't trying to say "Rider is right," but instead "Saber is wrong." (Just like everyone else.)

Second, this is simply a part of the larger critique on complete self-sacrifice ignoring one's own wellbeing and trying to take the whole burden of things onto yourself. "How to run a country" is by probably the largest-scale of the arenas this discussion plays out in, so, you have to take that into account and note that Fate is talking also about smaller-scale things like personal relationships.

"Saved but didn't inspire" is basically supposed to refer to the "taking the whole burden onto yourself" bit. Saber tried to do everything herself in order to save her people, but that meant that she didn't adequately rally and rely on them in turn. She saved them by being the lone hero, rather than by rallying them defend themselves - and ultimately failed. Saving people isn't a problem by itself, but the point is that it isn't necessarily a complete solution either. See also: Shirou's failures when trying to do this same thing.

"Nobody wants to be a martyr" is talking about the damage of self-sacrifice, in two ways. One, Fate is pretty clear on saying Saber had a shit life and didn't enjoy it either, and also that other characters trying to do this had similar experience. Two, it points out that self-sacrifice is something that Saber tries to do herself, but wouldn't want other people to do. This doesn't exactly work out, since other people will still follow the path you lay; F/Z got destroyed by trying to hold himself to the standard Saber set, even when she specifically did that so others wouldn't have to.

"No regrets" is a little different. It's also a theme in Fate, but it shapeshifts a lot especially if you identify it somewhat with "sticking to your ideals", and it also gets shut down a lot as well. I would say that that particular point isn't necessarily being held out as exactly correct; instead, Rider is just being given the chance to add a version of it to the some of the others within Fate.

So, the main points being argued here aren't that a king should live only for themselves, nor that a king shouldn't try to save their state, nor that a king shouldn't be righteous. Things like "A king should be greedy, a king should embody the extremes" are just Rider's particular point of view, being used as examples of alternatives to self-sacrifice. It's trying to point out ways in which utter self-sacrifice isn't actually the best policy, even if you're a strong enough person to see it through.