It is actually, as far as I know, a bit of a myth that babies 'used to always sleep on their fronts' as people often say. Up to about the 1950s it was advised to put babies to sleep on their back or side. The advice seems to have changed fairly suddenly. My grandmother was advised to put babies to sleep on their back, but by the time I was born (am in my mid 30s) my mother was told to put me on my front.
The change seems to have come about for a number of reasons, mostly to do with babies sleeping more deeply on their front and also worries about choking if they were to vomit. Other reasons included strengthening the neck and spine. One source I found says that Dr Spock's books changed from advocating back to front between 1955 and 1956 editions and that by 1958 he was strongly pushing front sleeping (though I don't have copies to check).
To me, personally, it has never made sense to say that babies were routinely and instinctively put to sleep on their fronts. As you lower a very young baby, you hold their neck, which makes it as least as natural to lay them on their back. Also, for most of human history, most babies co-slept. Which wouldn't allow for front sleeping.
So actually, the window of front sleeping as standard (as opposed to because some mothers found it suited their child) was pretty brief. About 40-50 years. Mind you, during that window my mother was berated for putting me on my back to sleep even though I screamed the house down every time she tried my front 
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My babies have hated tummy time. With DD1 I diligently tried. With DD2 I hardly bothered. With DS (newborn) I will do the same unless he actively likes it. He's a baby, I've no urge to train his stamina at something he dislikes. They spend a lot of time in slings, being carried, etc. Both DD's were also sitting by 5 months and spent a lot of time sitting up. They were late crawlers, but walked bang on average (and judging by a video I found recently, at least DD1 was running very fast about three days later).