I also think checking price before buying Bitcoin goes against DCA strategy.
Checking price before buying is a mandatory step if you don't set up a bot that automatically buy bitcoin for you and does not care about price. If you do your DCA manually, you have to check price. Want it or not, you must check price or see price when you are making your order.
DCA means investing fixed amount of money regularly no matter price. It is about investing automatically and not making emotional decisions based on market ups and downs.
Nothing is wrong in your clarification but if price during a single market cycle, already assumed doubles, it's risky to make your DCA at that time. It's better to wait for correction to buy bitcoin and in the meantime, you can prepare your capital for DCA.
If your plan is DCA weekly with $100, but if you have to wait for correction like 4 weeks, you will make a purchase at $400 when you do it with a correction. It's not bad in my opinion and I am quite sure experienced investors will wait for corrections to accumulate cheap bitcoin.
Or you can do both. One part is DCA no matter of price, just buy it weekly. A second part is waiting for correction and buy more. You are combining DCA and smart DCA.