Non driving parent expectations

I’m quite close to retiring but holding out because I’d lose my excuse for not being available to DM at all times.

If you're a nervous, anxious driver then that isn't safe for anyone, so I never criticise people who give up driving. But I agree that you shouldn't then just expect others to drive you whenever you want. You need to arrange your life appropriately.
This is essentially me, but I live in a city with good public transport. DH will drive us places but I never expect and always appreciate it.

That's true to a certain extent, but it;s also true that people need to start thinking about how they will be able to manage when they can no longer drive due to ill health / bad eyesight, unaffordable insurance etc.

IMO we have far too many dangerous elderly drivers on the roads these days. Using bus passes, finding a reliable local taxi service, online deliveries and relocating if necessary.

My DM has never driven and had no DH/DP to drive for the past 40 years. Public transport and (very reluctant occasional) taxis.
It's by no means an impossible life, it's the change that's difficult for your DM

This is no help to you OP but you have highlighted how important it is to KEEP DRIVING. My DM gave up driving when she met my stepfather (she was only 47!) and now she’s too anxious.

It’s a lesson for life really - being reliant on someone else to perform any basic skill makes you vulnerable and needy. Which is a horrible place to be.

Now is the time to say no. Your DM is behaving exactly as my MIL did.

It was definitely the company / emotional support she was looking for. She had never done any of that stuff - hospital, funerals etc without FIL.

I think ‘Can’t get time off work’ is the easiest way, and when it comes up again just say that it’s been noticed you have had a lot of time off and you are at your limit.

Online grocery shop and taxi numbers plugged into her phone, or Uber if she has it and can use the app.

It only gets worse if you don’t say no.

Probably the best strategy is to get in first and occupy the 'useless/unreliable' position. I realise that option is now taken by the 'flat out refusal' person in the dynamic OP.

You are really so good already. Put your foot down or it will continue. She will find another neighbour to take her I'm sure. This might be about having you at her beck and call for company rather than really needing you.

Alternative to online shopping or going to the store, use “Milk and More”. They have a website and have a wide variety of groceries, all delivered to you DM front door.
For getting about, your DM can either use public transport or get a taxi. If her doctors is only a ten minute walk it won’t cost very much. Encourage her to be as independent as possible. The more you run around after your DM, the more she will expect it and your time with her is spent running her around instead of being spent enjoying each other’s company doing things you would both enjoy.

Well the sibling needs to help with groceries either by doing an online shop for her taking her list via phone or teaching her how to do it so neither of you has to. Or taking her. If they refuse all of that you point out how selfish they are in expecting you to do it all and you bat it back to your mum and say that you will do alternate weeks (if you are happy with that) and she needs to either make her own arrangements or get 2 weeks worth of shopping.

Regarding the funeral you just say "No". Surely she knows others who are going that she could ask or she has the money for a taxi. Your employment is more important than getting to a funeral.

I get it, I really do, I had a demanding mother who didn't drive, a demanding job and a useless brother. There was a helpful village volunteer service and helpful neighbours and mum had enough money to pay people to help her regardless (and I offered to help her find someone trustworthy who could help). But no, she expected me to do it all. I did start saying no eventually but it nearly cost me my mental health and it irreparably damaged my relationship with her.

Online food shopping and taxis need to be in her life.

Maybe tell her you can't get the time off but you'll organise a taxi for her. She won't like it but what can you do if work "won't let you take the time off".