I'm not tone deaf to how many people are losing jobs and having a hard time right now. I don't want to be announced as getting a promotion plus pay raise when others are struggling.
There is some reason to your interpretation, but counterpoints can be made. For one, you enabled working from home for several teams, which means the company can remain operational, which is to the benefit of keeping people employed and paid. That both justifies your promotion and raises spirits in regards to future employment.
If played right, this actually shows employees that keeping up productivity will be rewarded, which is beneficial to employee morale and the company.
You do indeed need to handle the optics, and an announcement could be deferred if the current atmosphere is gloomy bordering on toxic (but that's a whole different ballgame in and of itself), but it's not as bad as you're interpreting it to be.
Some (less professional) colleagues made disparaging remarks about me due to my enabling them to work remotely (guess some of them wanted a free "stay-cation" at home; they don't seem to realize I likely saved their jobs)
Pointing out you saved their jobs is the best response to this, and don't engage any further. Don't let the bad work ethics of others hold you back, that makes no sense.
One individual told me he "would have preferred 7 months' severance than his job".
In an informal capacity, I would ask them if they want me to share that sentiment with their manager. At worst, nothing comes of it; at best, they terminate him anyway. It's a supposed win-win for him, right?
Not to be uttered as a threat, but rather as a way to get them to reevaluate what they're saying (and treating you based on that) and what thay says about their value as an employee. Don't actually raise that point to their manager. Unless maybe they explicitly tell you to, I guess, but that's up to you whether you want to be the bellringer (I wouldn't).
Some teams are actively being laid off, with a heavy focus on software teams that are idling right now (not many, but they exist).
This does contribute negatively to the optics; but again it also suggests that your team is still operational and will remain so, which raises the morale of the employees who (sad to say) will remain employed; which is a net positive for the company.
It may be actually beneficial for your team to announce this internally (to the team), but it is likely to cause some blowback in other teams. This is an optics game, but it's not just your responsibility, it's that of your employer. Do raise your points with them, as it is constructive; but I wouldn't go so far as to tell them how to handle it.
Some teams were effectively automated due to my technical implementations (legacy software and DevOps teams), and are also being laid off.
It's a common interpretation that automation "takes away jobs", but that simply isn't the case. There are more jobs now (with higher work quality) than there were before the industrial era, and every automation advancement we've made since. Computers automated the calculator jobs (people whose jobs it literally was to add numbers) but gave birth to multiple industries at the same time.
The ones clamoring for taking away jobs are the ones who are unwilling to adapt to changes in the industry. Where one jobs closes (due to automation), others tend to open (e.g. automation implementation and maintenance).
Furthermore, it's an IT professional's literal job to automate things. If you hadn't done your job, you would've been the one without a job. You shouldn't hold back your own career advancement for that of others (but don't go over corpses either, of course).
Some members have directly contacted my via personal e-mail to say nasty things to me.
Depending on how nasty and recurring these comments are, either ignore them or escalate it to your employer. It's not going to do their work reference any good to lash out at coworkers, and you can get some protection against this if it grows beyond one or two nasty emails.
Is it reasonable to request additional discretion for a major promotion?
It is reasonable to ask the company to consider the optics and potential consequences to your workplace interactions.
It is, in my opinion, not reasonable to tell the company how to handle their own internal business.
Is it a bad idea to request this (i.e. appear to be second-guessing my supervisor's judgement)?
Any manager who is incapable of listening to an outside view (that does not outright state that it's more correct) is not a good manager, and generally does not make for an enjoyable workplace anyway.
My personal stance on that is that I wouldn't want to work in a place like that anyway, if they effectively slam you for trying to constructively contribute.
My concern is more about the morale and sensitivities of my colleagues rather than my public appearance/impression (friends at work are still good friends, and I can shrug off the nasty comments I've received easily),
As I said, I agree that the optics here need to be considered; but I do think that the biggest part of that lies with your company, not you personally.
Definitely feel free to raise a genuine concern with your employer and see if you can work together to find the best approach, but don't just cut into your flesh (avoiding the promotion) because of some nasty comments from others.
though admittedly I care most about my income (and ability to provide for my family).
Opinions on capitalism aside, your stance is perfectly reasonable. The ones who've been making nasty comments wouldn't think twice about reconsidering something that benefits them personally, why should you?
As long as you don't take the pride road on this promotion, which you clearly aren't, you shouldn't be blamed for wanting a promotion that you're being given.