Sounds like a lovely plan, @Wadingthroughshit, & if you follow some of the suggestions here have no doubt you can find a suitable new family friend ...
However, if this is going to be your first dog ...? - may I politely offer a couple of words of caution, which I genuinely don't mean to come over as patronising or scolding?
GARDEN
Yes your dog will need & enjoy your gardens, but make sure at least one of them is 100% secure. Dogs are great at escaping & will follow exciting scents through hedges - & some of them can jump far higher than you may imagine.
PODENCO/GREYHOUND
Gorgeous dogs & lovely natures but please beware of their exceptional prey-drive. Highly likely to chase cats/squirrels/livestock & this is not a trait anyone but a full-time expert could hope to train out. Unsuitable around small kids for this reason - you can imagine the conseqences of dog dashing over roads etc or even on lead jerking you/kids off your feet to chase perceived prey.
TRAINING
You come over as measured & thoughtful so please excuse this one - but unless you can make a regular once-or-twice weekly timeslot for this & stick to it very long term, maybe rethink or postpone the plan?
It needs to be a fun activity for you & the kids, with an experienced APDT-recognised trainer. The trainer is there to train you & ESPECIALLY the kids in how to guide, manage & control your new pal.
Outside of those formal classes, 'training' has to take place every day - in the same way that we are constantly monitoring & iyswim 'stealth-training' our children. It takes patience, consistency, kindness & stamina ... & if you are happy with that, these are great things to instill in your children.
ANXIETY ISSUES
Gruelling for you & your child. Totally agree that having a dog to love & be loved by can absolutely assist with this.
I hope this works out for you & you all get to enjoy the fun, learning, & emotional security that comes with welcoming a dog into your family!