Sorry your DF is so unwell, it's a very hard time for everyone in the family.
Your DB is clearly lashing out and should not have used that kind of language, but I think it sounds like he has been left with the bulk of the work with very little assistance. That would have been hard for him- and hurtful.
I understand that your DF asked you not to visit, but that doesn't mean you couldn't have offered other assistance. Your DB may be in charge of wills but that doesn't mean he has to do all the work supporting DM and finding DF new nursing home etc. Perhaps he could have asked for help, but people often feel they can't/shouldn't have too.
Other than offering lifts, sometimes dropping off shopping and pads, have you done anything else to help? If not, it doesn't really seem all that much, in comparison to the leg work it takes to find a new nursing home (never mind the emotional aspect- worrying if you have made the right choice, will DF be as content as can be expected there etc).
I think messaging him was also not ideal- you don't always get the tone of what is meant and it can come across differently to the intent. Plus, I think that kind of conversation should be done in person or via phone call where you actually SPEAK. I use text messaging as much as anyone, but not for important or emotional conversations. It's so impersonal.
Unless you live a long distance away, there isn't really much excuse and I can see why your DB is upset. He should not have spoken to you like that, it's never ok to use language like that- but you are in the wrong too.
Also what does the fact he is educated and wealthy make to the situation? He's still losing his dad, which I've no doubt he finds as hard as you are. Wealth doesn't change that.