Op l had 2 dcs 16 months apart, and a 6 yr old Weimaraner, my DH worked away a lot so was rarely home to help...
From when my dcs were 8 weeks old, l took both dcs in a huge double buggy out with our dog, at 7am all year round..for an hours march through the woods......it was also my fitness routine!
As they got older l let first one, then both, dcs walk part way but always had to put them in the buggy on the way back as they got tired. We used to repeat this for 30-45 minutes in the afternoon..
It is perfectly possible, but you have to really be willing to do it every day and commit to it, for me it is worth it as life without a dog is miserable for me.
I also drove to GPs, 4 hours away, 3 times a year on my own with DCs and dog, it just takes good organisation and a bit of extra planning.
Most dogs will adjust to being somewhere else when you go on holiday, but budget on a house sitter who has experience of dogs, in case your dog hates kennels as some do. And you need to wait until they are a year old before leaving them for any length of time.
If you are careful to buy direct from a reputable breeder often they will take them whilst you are away. Think about a lower energy breed if you are daunted about the amount of runs a dog needs, a walk on lead even three times a day, is not enough for most breeds and you need to think where nearby is secure to give your dog a run and play games, they are only a puppy for a short time, its the long haul you need to consider.
When my Dcs are sick, and l cannot walk out, l use a dog daycare collection service, but its not cheap, ultimately you need to have a hefty budget to cover all these times you may need help.
Depends entirely where you live so research this.