I have done both. I did savings bonds for DD starting back in the 80's. I don't think the return was very good compared to a diversified investment. I would not do that today with today's interest rates. Besides, it was a pain when it came time to redeem them. We had to sign each one and fill out some information on the back. I am not sure if they are virtual these days or if you get an actual piece of paper when you buy the bond.
With DS1 I went the UGMA mutual fund route and both the returns and redemption were much easier.
For DS2 & 3, instead of saving and investing a separate stash, I included the savings in my 401k investments. Advantage: the money is mine, for DD and DS1 the money was in their names, they used it to go to college until the money ran out then they quit college.
For grandchildren I would go with a 529 plan. Advantage: if the grandchild does not go to college the money can be used for another child. You retain some control over the money. Plus, depending on the plan you may have a variety of investment options.