Way more important than savings rate vs. mortgage interest is mortgage interest vs. inflation (at least in the US, where the savings rate is practically 0%). It's probably one of the best times in history to get a fixed-rate mortgage (and pretty much any cheap debt you can get your hands on) in the US, but that doesn't mean it's a good time to get or keep/refinance a mortgage for you. Inflation/Fed rates have been astoundingly low, which has to eventually change, likely quite significantly where your debt is devaluing significantly faster than the interest rates you can lock into today with you having access to savings rates >0%. Of course, right now, you could use all these different "accelerated checking" schemes (credit unions in particular are good for this, but they're always named something different) which actually pay decent interest (~5% annually) if you jump through a bunch of hoops (generally, something like ACH deposit 1-3 times [Paypal] and use your debit card 5-12 times [Amazon Prime membership will do you well, here, though Meritline can be a great choice, too]), and they usually have maximum deposit limits of $5k-25k.
Most mortgage balances are much higher then this so if you wanted to keep the entire amount in "checking" then you would have a much lower effective interest rate on the money you could use to payoff your mortgage with. It is also important to understand that these interest rates on checking deposits are not guaranteed to last forever, while you are guaranteed to be due for a set amount of interest (based on the outstanding unpaid principle) until you pay off your loan.
Another point is that if you were to pay off your mortgage then you can get a guaranteed return on this money over 30 years (you no longer need to pay interest on the money you use to pay off your debt). On the other hand, if you were to invest in anything but treasury bonds, you would be taking on some level of risk and your return would not be guaranteed. When comparing paying off your mortgage verses investing in treasury bonds, paying off your mortgage would be a better investment.
Finally if you were to pay off your mortgage, you could then use the money you would use as a mortgage payment to invest in something over time, using dollar cost averaging, preventing you from investing all of your money at the market top.
Well, you can have more than one bank/CU account open
-- but yeah, the more you have, the more time you'll have to spend meeting their hoop-jumping requirements, but this shouldn't be more than an hour per month per account (and this could be scripted if more than a couple accounts are needed to be open). While interest rates are at historical lows, checking/savings rates have virtually nowhere to go but up, which is why the fixed rate available in HELoCs are so attractive. I'm not interested in them, ATM -- I don't want so much at stake and would be surprised if I could get a HELoC at a good rate, but I'm always on the search for 0% credit card promos.