TLDR: is it worth buying the house I want to retire in for more than I currently pay in rent, or should I buy a less ideal house for less than my current rent and upgrade later?
No. Neither.
Everybody buys the house that they're going to retire in, but I've only met one homeowner who's actually done it.
Everybody wants to buy a "good starter home" and upgrade later, but if the current apartment suits your needs (at less than the cost of ownership) then why buy an intermediate house with an extra set of transaction costs?
I'd suggest that you keep looking at neighborhoods and homes, and keep going to open houses. Wait a year or three. One day the right home will come on the market (probably in November or January), and after you've been through 40-50 open houses you'll know exactly what you're looking for. If it's way below the cost of renting then you could buy it, but only if you're putting down at least 20% and the math (including closing costs, maintenance, and all the other fudge factors) is overwhelmingly in your favor. You don't want to leave your current apartment unless your circumstances change (for example, you start your family with triplets) and you definitely don't want to move to a nasty commute just for the sake of a nice house-- which you'll never see because of that nasty commute.
While you're looking at houses, you could practice your homeowner budget. Figure out your ideal price (the one that's cheaper than renting) and then see if you can live within that budget. Then see if you can still live within that budget assuming one of you gets laid off or decides to stay home with the triplets.
Don't get pushed around by the housing market. You're not trying to buy every house on the market, so you don't care about prices going up 10%. You're only trying to buy the place that's right for you, at the right price, and perhaps at the right time. No matter how high the market gets, you'll find bargains. Eventually the market will retract (especially during winter) and you'll find more bargains.
Many years ago, Elizabeth Warren wrote the book "The Two-Income Trap" about dual-salary couples buying high-priced real estate... and then one (or both) of them got laid off. While you're checking listings and visiting open houses, it's worth digging that book out of the library and reading it. I'd also recommend the book already mentioned on this forum, "Rent vs Own" by Jane Hodges.