Three months seems like an unusually large severance to me. And I've never heard of an employer paying for COBRA at all.
Many employees never negotiate these things, sadly, because organizations try to create a taboo about talking about what would happen if your tenure with them ever comes to an end. As a result, a lot of people are made to accept no severance at all, or extremely poor severance, like 3 months of salary, if any.
In all cases, severance was never offered, but was immediately agreed to when I countered the original offer and asked. Support for on-going insurance has varied, and the specific amounts have varied, but I can say for sure I would reject a company immediately if they were unwilling to offer more than 3 months of severance pay in addition to extended support for, at the very least, on-going health insurance.
From my personal experience, I just try to be very direct and straightforward with my understanding of the risks involved in taking a new position.
If a position requires me to relocate, then what happens if the company simply decides to let me go for an arbitrary reason shortly after I have moved? I am now stuck in a place without a job, and may have to incur costs to relocate to wherever I was prior to taking that job. This actually happened to a friend of mine who left a job in London and told his landlord he was leaving -- he moved to San Francisco (on a pretty low relocation benefit) to join a start-up and literally the week that he arrived they said they were restructuring, no longer had the budget for his position, and didn't want to sponsor his visa for other positions that he wasn't directly qualified for. He suddenly had to get out of his lease in SF and figure out whether he wanted to scramble to find a visa-sponsor and stay or go back to London. He ended up going back to London, and ended up on net losing money on the overall experience by having to ship things back to himself and pay a new fee to a realtor to help him find a new apartment. Severance is a mechanism by which you say that if the company wants to hire you (they may or may not) then they have to prevent that kind of outcome by guaranteeing they'll compensate you if they put you in a position to scramble.
That's an extreme case, but the idea is the basis for asking for severance. You simply say that in order to feel comfortable relocating / leaving the old position / joining a risky company / joining a crowded industry or market where other companies have recently had layoffs / etc., then what it takes is for you to see a gesture of commitment from the hiring company in the form of a severance benefit.
I try to offer to be flexible about the terms and also tell them that terms which offer a continuation of health insurance are the most valuable for me.
I typically try to delay talking about severance until all of the other properties of the job offer are agreed upon, because it's a very different issue than salary, bonus, equity, vacation, or perks like remote working. If those things aren't mutually satisfactory, then there's no point in even bringing up severance at all.
I also try to delay talking about relocation until the latter stages too, for similar reasons, and I find that talking about severance in unison with relocation, and explaining the link between the risk of relocating and possibly being stuck scrambling if there is a lay-off and the need for severance is generally well-received.
You should be prepared with what kind of severance package you feel is reasonable, and possibly some examples of other companies that have provided it. It should be a written part of your offer letter, including specifics about insurance continuation, and you should be given the chance to review those details before giving a definitive yes or no answer. Be confident when you ask for what you're worth, and in the case of severance just remember that 6 months of salary, even for a highly-paid employee, is not a significant extra hiring expense for most companies. What you're asking for is not some outrageous or greedy amount -- it certainly won't shower you with riches or materially change your life, especially if you actually do fall on hard times and have to search for the next job for a long time. It's merely a reasonable amount of money to protect you from a risk. The hiring company does not have to offer that protection, but if you value yourself (and you should) then you likewise don't have to work for any company that would choose not to do it.
I'd say the minimum I would consider reasonable would be 6 months, maybe more in higher cost of living areas. That's just the cost of doing business.
If you relocate for a job, or give up a stable current job, or a number of other things, you're just essentially trusting the new company not to do this to you, and often it's completely blind trust and you have no way to know in advance if your trust is well-placed.
If they are willing to compensate you for a situation where they unexpectedly (not due to any cause by you, like poor performance or insubordination or something) throw you into unemployment, possibly in a new location where you lack a professional network to find the next job, or where you would not choose to stay apart for the job you took, it shows a willingness on their part to earn your trust.
It's extremely bizarre to me to hear someone characterize taking a new job by saying "I don't understand what the risk is." It's a huge, huge risk every time that the new place is going to treat you badly and that you could not have seen it ahead of time because they didn't emit red flags during the hiring process. And many ways of them erratically treating you badly results in you suddenly having no income or insurance and needing to figure out what to do -- a situation that should absolutely never be financed out of your personal savings (though, often it is because people rarely negotiate adequate severance).
Perhaps you've been very lucky with working for good companies and just haven't experienced how prevalent this is.