Sounds like you might be lucky and just need a new screw for the Milwaukee. Hope so.
I use a cordless Makita 1/2" impact wrench for that kind of work. Much better tool than a straight drill.
Also, there are better structural screws available. The Spax and GRK type structural screws do not need a pilot hole and they go in faster than an old fashioned lag bolt. By the time you drill your pilot holes, you would be done with the whole job using modern structural screws.
Another point is that, depending on the wood, large old fashioned lags actually might need two pilots. One smaller and deeper hole where the threads are, and one larger and shallow hole where the shank is. Mostly it doesn't matter but on hardwood and wood that wants to split, drilling a shallow larger hole for the shank makes a lot of difference.
Finally, when I had a lot of those big old time lags to install, I dipped them in a can of motor oil first. Makes life much easier.