I agree that if bitcoin gains more popularity, it'll likely pull the DOGE up. Simply, if someone put some effort to learn about bitcoin, than he/she is very likely to try the other cryptos as well, and DOGE is one of the most popular and easier to get.
Don't really see what's the strenght of LTC nowadays tho. Bitcoin is (was and will be) more popular as both, functional currency or investment. Dogecoin has low tx fees, creates more hype and reaches to a new people. Think LTC will be slowly losing its position.
I'd think of it less as "Why is Litecoin so high?" and more "Why is Dogecoin so low?"
Even if anything does happen to Litecoin in time, we're merely adding a mechanism to get their hashrate as well as our own, helping secure the coin, we can still survive without them.