First of all I would like to point out that apart from する, なさる and できる and other する-ish verbs are also game.
Second of all, it is important to note that ～たり～たり is a 終止形 construction and thus supposedly forms a bracket of a complete idea or sentence. 終止形 usually doesn't interact with anything specifically outside of itself, but when it does it tends to act like a nominalization that can at times form compound nouns and at other times like a quotation bracket.
My classical dictionary actually mentions this nominalization as an argument for why たり is paired with する. The very similar construction ～ぬ～ぬ is -not-, and the logic behind this is that it is not volitional, so it conflicts with する, and ends up just attaching to a verb that describes its action: "浮きぬ沈みぬ揺られければ"
There are several examples in my dictionaries that have ～たり～たり and ～つ～つ (which たり is derived from) that are not directly connected with する, but instead they are succeeded by a noun phrase that is equivalent to their action:
"僧都（そうづ）乗っては下りつ、下りては乗っつ、あらまし事をぞし給ひける" == 僧都は乗っては降りたり、降りては乗ったり、乗りたがっている事をしました。
"掃いたり抜（ぬご）うたり、塵拾い、手づから掃除せられけり" == 掃いたり抜いだり、塵拾い、自分の手で掃除しました。
However, the volitional nuance for つ/たり is completely lost in modern Japanese as ぬ is completely gone, so it's not worth going into any further.
Now, moving on, your first two examples have serious problems.
I searched "立ったり座ったり働く" and got "立ったり、座ったり、働くときの姿勢..." - This is perfectly correct. Looking at the former phrase, it is disjointed because there is nothing connected between 立ったり／座ったり／働く (it would be "I stood, I sat, I work" in an archaic sounding Japanese), but as you can see in the latter phrase, this is exactly how it's supposed to be. In other words, the parts of the phrase interact like this: 立つこと、座ること、（などの）働くときの姿勢...
I searched "話したり笑ったり食べる" and got "自信を持って話したり、笑ったり、食べる事が出来る". - This is again perfectly valid, if not a little strange. It equates talking and laughing with the act of eating, but I suppose in a way that makes sense. After all it's not like anyone needs confidence to eat. (Extracting the たりｓ gives you 自身を持って食べる事が出来る.) It is in regards to the act of eating.
First of all 可愛かったり、美しい人 is not really equivalent to 可愛かったり美しかったりする人. I would assume it is closer to: 可愛かったり、いや、取り敢えず美しい人. It's more nuance than anything else, especially in this case, but without する, 可愛かったり is properly disjointed from 美しい. That is my academic argument, but the realistic one is that because saying something like 可愛かったり美しかったりする人, or 可愛かったり綺麗だったりするもの is just too much of a mouthful for some people. It probably just got なきゃ'd from it's original なければならない. Again, this is partially possible because without much of a stretch, it still makes sense grammatically but with a different but not quite as important nuance.
I again use my former argument: "だるかったり、(取り敢えず)キツい日" キツい irregardless of whether the reason for that is it being だるい or not. - I did not find any examples of this on Google though.