I was shocked when I found out bounce houses cost ca. $250 to rent (FELLOW MUSTACHIANS - LUCRATIVE SIDE HUSTLE OPPORTUNITY). My antimustachian BIL once got TWO for his kids' birthday party. His reason? That way kids wouldn't get bored. At those rates, it makes sense to buy a $150 version and use it for fun in the future, or sell it on Craigslist "nearly new" after the party.
I agree that home parties can be very Antimustachian, especially if you provide tons and tons of food. For this reason, we have opted like you to do parties out of the house. $5 a kid is likely cheaper than providing lunch for parents and adults at your home, especially if you feel like you need to provide alcohol.
The scales tip the other way once kids are older, since it is unlikely that a parent of an 8 year old will want to stay at your house. Then you are only providing food and cake for the kids.
I think at some level, celebrations for life events are not, by their very nature, frugal events. They can be, but often they are not. This is one reason that many on here recommend only having family parties most years and limiting kid parties to big birthdays - like 5, 10, and 16, or something like that.
I went to a party last year for a kid turning six. It was at a train station (rental $200). They also had entertainment (jugglers/fire eaters). Who knows what that cost. And this is where it gets REALLY expensive/awesome. They had super fancy food for the adults. Catered pizza with arugula, dolmades, etc. in chafing dishes. That was a kick ass party for the adults especially. I left thinking that they had dropped at least a grand on the party (not counting presents, which they didn't open). I learned a year later that the father is a local chef and probably got the food free from a friend (who had a restaurant down the block) and might have cooked some himself.
Is your kids' birthday in Fall/summer or spring? Renting a pavilion is usually pretty cheap.