OP, from the negative tenor of your question, the way you've blasted it in a lot of Asky subreddits, and your general post history, I think you're working through some things about yourself, what it may mean to live in a body without testosterone, and if historical eunuchs are your spiritual ancestors. I wasn't going to engage, but you know what, there's probably less than 5 people on the internet who could meaningfully talk to you about this, and you’re clearly in distress, so let's talk.
So the vibe I'm getting from your question is basically ... A) Is lack of testosterone a disability? B) If so, why were these people in history intentionally disabled by themselves or others? And furthermore, assuming A and B, why did society explicitly make and value these disabled people, when society traditionally does not value disability? (And the subtext, if I do not have testosterone am I disabled, am I valued by society? Which I have firm opinions on but is outside my scope of practice, and you'll have to figure out for yourself.)
Let's tackle your assumptions first. Assumption A: Periods in history where eunuchs are "bodyguards, soldiers, or assassins." Those are some jobs you've thrown together that perhaps shouldn't be. Bodyguards - yes, eunuchs were often bodyguards, but more in the sense of an honor guard. I didn’t expect the geriatric veterans firing blanks at my 92 year old grandpa’s funeral to be eunuchs, but that is way closer to what role a eunuch held than the Secret Service. The bodyguard role was (to grit my teeth and be extremely reductionist for the sake of time) symbolic and often religious or quasi religious. Assassins? From time to time a eunuch would do a poisoning, but that wasn't a full time gig. If they did that, it would be because their real role as a close body servant gave them access and trust, not because they were ninjas zipping up windows. Soldiers? No. The idea of eunuch armies is popular from a certain book/tv show that I have never seen and yet am cursed to answer questions about. Eunuchs, when they were in the military, were in high ranking positions, usually given to them by rulers. They did not work up from letter carrier to postmaster general, they went straight to the top. Think Narses, not Unsullied.
Assumption B: Eunuchs as disabled, in particular you say: "frail, less decisive, hampered in your cognitive abilities, and more prone to mental breakdowns." There's a lot of work you're putting on old T here. Frail? Eunuchs were certainly prone to osteoporosis, so their bones would weaken over time. However, pre-pubescently castrated eunuchs seemed to experience it differently, compared to contemporary old people, the bone density of an entire castrato skeleton studied in 2013-16 was different than expected for senile osteoporosis. (look at figure 8) The authors conclude he had made "a long-standing adaptation," aka, learned to live with it. I assume you also are thinking of lower muscle mass, that’s harder to study on skeletons, but the article does talk about very particular muscle development for a professional opera singer and the markers on his bones for that. So he was absolutely able in muscular ability to do his job, an occupational therapist would pronounce him fit for work, however we do not know what our friend Gasparo could bench, alas.
Now, mental abilities. Hormones for sure do something on our brains, but the what, why, and how much are all in debate. There is decent evidence low testosterone contributes to mild depression in older men, and possibly dementia. It’s a hard thing to study. There are no controls, they’re studying prostate cancer patients, the elderly, or various specific endocrine disorders. No one’s doing a double blind castration challenge on human subjects. There is also a big difference between growing up with a certain brain-body connection and hormonal profile, your neurons pruning under those hormones, vs having your hormonal profile change along with your brain aging. Emotion and mental health history is also very hard, because the way we relate to our minds is so different after 300 years, even within the same culture. But much like the birdbones, historical eunuchs may have just been used to it and already mentally adjusted to any depressive tendencies. I can tell you many eunuchs lived good lives, worked opera well into their 60s, and enjoyed an active, social retirement worthy of any assisted living brochure.
General low mental ability and indecisiveness? Well they have a similar hormonal profile to postmenopausal cis women, so I invite you to go down to your local quilt shop, posit this theory, and see what happens to you. Cognitive impairment is an observed, but hard to quantify side effect with prostate cancer patients. It is mild enough that they can still go about their existing lives. (I am sure we could find a case study saying someone was super hampered, but I am only checking up on systematic reviews today.) Again, it’s hard to say how this applies to historical eunuchs, since there is a big difference between growing up with your eunuch brain vs getting power-blasted with hormonal blockers at 64 AND you have cancer. As a counter, eunuchs were usually stereotyped as conniving and intelligent in their societies, but I think that was probably because they were politicians and learned to read and stuff. They were certainly put in positions requiring great intelligence, like accounting and military strategy. I know of a castrato who was into investing, which everyone thinks is very intelligent these days. They probably mentally did as well as any other average Zhou in similar circumstances.
So in conclusion, simple testes-only castration was a mildly disabling condition in life without modern medicine, and probably way better than having gout or rickets or fistulas or other such common complaints of 18th century Europeans. They still held down jobs and lived good lives. (Complete castration, no penis no testes, was way worse for your health, but that does not have to do with testosterone, it has to do with having a stoma entrance to your bladder and the resultant infections.)
So why weren’t they soldiers? Not because of their bodies, but because they were way too expensive. In societies where eunuchs were enslaved, and where they practiced complete castration, a eunuch was worth about 10 times the price of a normal boy. Now, the idea of “cannon fodder” and disposable soldiers is wrong, or so I have frequently read on AskHistorians, but if you’re looking to assemble a unit of 100 people, would you rather spend $1,000, or $10,000? You’re not spending 10K unless it gets you something amazing, and eunuchs did get you something amazing, but only in very specific contexts, which was not battlefields. Eunuchs were high value human capital, to use modern HR terms, and you deployed them only when you absolutely had to, which was only in certain spaces such as religious venues and around royalty. The very last eunuch guards were (are? They are hard to keep tabs on in English) in Mecca and Medina. For why they were so valued in these spaces, please take a look at the second list /u/gynnis-scholasticus has so kindly assembled.
Back to today, and you. Eunuch is a gender identity some modern people use, so if you feel called to embrace it, you're not alone, and it is valid, even if you’ve never met anyone who identified as such. It is somewhat diverse, with some people who simply seek castration surgery, and then once they have it are satisfied and do nothing else, some cancer patients, some people closer to transmen. Some modern eunuchs also find great meaning in having a historical heritage to their gender identity, namely Richard Wassersug, who is the leading academic in modern eunuchry, and he identifies as a eunuch. Here is his Google Scholar profile. (he also knows a lot about frogs.) And here is a nice popular article from Dr. Wassersug: Embracing a Eunuch Identity. Don’t let your historical stereotypes about eunuchs and the perceived superiority of testosterone stop you from being who you want to be in the body you want to have.
As you are waiting you can look at some earlier answers on here on the subject of eunuchs, written by the amazing u/caffarelli!
On the physical side of being a eunuch:
On the reasons eunuchs were used/employed:
I hope this is helpful, or otherwise that you will get a new response from one of our users! I have not seen Caffarelli here in a long while, but we have some other users who know much about sex and gender in history and should know a bit about eunuchs. Cheers!