Sighthound specific rescues do get litters of pups in, they just tend not to go on the websites, as they already have pre-approved homes waiting. If you are willing to wait, you can usually register interest and fill in pre-adoption forms, so you’re vetted/approved when they do have pups come in.
Both Lurcher Link and Evesham Greyhound and Lurcher Rescue consider homes on an individual basis, so there’s no strict age limit for children and they will place dogs with children where appropriate. They also rehome nationally - under normal circumstances - but obviously Covid has caused problems with this, so you may have to wait until lockdown ends.
My boy is 15 now, but we took him on as a pup when ds2 was about 18 months old. My other Lurcher (sadly passed last Summer) came to us as a pup from a general rescue, rather than a Lurcher/Sighthound specific one, when dd was 4.
I adore Lurchers, but you do need to be aware that they can be very different according to the mix of breeds in particular dogs and that those bred from working lines will have high prey drives. It’s worth joining the FB groups for a few of the rescues to get some idea of what the different mixes can be like. Both mine were/are scruffies. My old lad is Deerhound/Saluki x Greyhound Bearded Collie and is the calmest, friendliest, most laid-back lad ever. We often say he’s the perfect family dog. We don’t know what our other boy’s breeding was, but he definitely had a heft dollop of Saluki in there, as well as some Bedlington, somewhere along the line, judging by his fur. He was extremely highly strung and very needy, but then he had an very bad start in life (dumped at a couple of hours old, so never had his mum or a litter to learn from, then very ill and missed his early socialisation) and struggled with some pretty serious health problems from a very young age, so a lot of his issues were probably down to that.
I do think Lurchers make fantastic family dogs, but it needs to be the right match and you need to be aware that, if you have little children, unless you are super careful, they are likely to be knocked over (Lurchered) a few times while the pup learns how to control his legs and braking system. All three of mine wore their Lurcher Link ‘I’ve Been Lurchered’ tshirts with pride when they were younger!
The good news is that if you go through a reputable rescue like the ones I’ve mentioned above, they will make sure you are a good match and also be there to support you all the way.