I am not going to comment on the "convenience" or "inconsistencies" on the various threads on this matter. I am kind of surprised that you have even bothered to ask, since you say that you work for local government and would presumably therefore have full pay for any peiod of sickness (unless, like my local government employer, they no longer pay anything more than SSP during a notice period, except for long term sick existing prior to notice being served or notice served by the employer - funnily a lot fewer people develop sickness during their notice period now).
However, as a disabled local government manager dealing with this specific situation, it seems that you have agreed with your manager to work from home for your notice period. The employer was not responsible for the chair you used working from home (surprising that as we would provide a chair for home working were required, but I guess things differ slightly between employers) - you bought it and were responsible for it. The fact that your son has broken is your responsibility and therefore you need to replace it and work. Your responsibility, your problem; just as you say that what is your employers responsibility is their problem. My son broke my chair is not sickness - it is you not meeting your own needs in much the same way as you say the employer didn't.
I assume that you are giving up work then, as if you are not replacing the chair then you won't be able to wfh / hybrid work for a new employer?
And I would require evidence that it was irreparably broken, at which point I would be discussing with HR whether you were on paid or unpaid leave.
In other words - you may be justifiably annoyed at your employers previous actions. I don't know enough about what happened and why to comment. But nobody is going to be stupid enough to believe the convenience of your story even if it were true - nobody here and nobody at your employers. They might let you get away with it. They might not. But that's a risk that you take. And it is a risk, because you might think that resigning draws a line under everything, but sometimes things catch up with you.
And if you are going to tell a lie, at least try to make it believeable. Why a fanciful stroty when you can just catch the cold virus everyone else has?
Oh, and I would dock te teenagers pocket money until you can afford a replacement / they have paid for it - they are old enough to act responsibily and also to recognise how important your special items are to you. There should be consequences....