squeaky - if you don't wear a seatbelt you can be flung around in an accident and kill anyone you come into contact with (whether they are strapped in or not).
I know how you feel OP, but I still regret some of the things I didn't let my MIL do, or stuff I moaned about...now that she is no longer with us. I agree your child is still quite young for essentially "unsupervised" trips...but if you can find a way to make it safe but let it happen please do.
I had a row with my MIL at DS's christening because she insisted on changing his nappy and I criticised her for the way she did it (poor eyesight led to poo not being wiped properly and thus the vaguest inkling of nappy rash). Why didn't I just change him again myself and not make a fuss?
I got cross with her on trips for spoiling him in shops as he could pretty much name whatever he wanted and she'd get it...(I thought that he shouldn't be learning that he could buy stuff every time). It took me a while to just think...let her enjoy him while she can.
My mum is 20 years younger than MIL was and also MY mum...so maybe I just trusted her more. She also didn't take quite the same huff about my rules. Nevertheless, I am grateful for the love my DS still has for MIL. She died when he was 4 and he's now nearly 9. We remind him of things they did together and she is very much included in family stories.
I wish you luck in dealing with this one. Attitudes change and we all do things differently but finding a way for older grandparents to still be involved is important (as I think you said you do try to do). Perhaps you should remind them that it's not really fair to promise your child something without checking with you and your DP first. Also remind them of all the things they'll have to remember if they want to have a successful day out. Think snacks, toilet breaks, changes of clothes...blah blah blah. Are they SURE they want the responsibility?! Then suggest something that will suit everyone.
However, don't compromise on safety! I think both sets of GPs thought I was off my head insisting on proper use of suncream...and even I have to admit I was a bit obsessed with moving the parasol on the buggy every time we turned a corner(!)...but DS has never had sunburn. They thought I was mad...but they got the message.