I towed my daughter on a tagalong for a few years. But I must say I didn't ever do it with my road bike and I don't think I ever would. I used my hybrid, which was a lot more forgiving of the movement out back (being a less flexible frame and heavier) and had nice fatter tyres and more substantial brakes.
Depending on the hitch system, where you locate it on the seatpost and the frame configuration, the loads can be close to point loads at the frame nodes, so they are not inherently bad. This is especially so on a bike which has been designed with some redundancy to allow for carrying loads (ie touring/hybrid bikes that you would expect to fit with racks and panniers and load on up).
But I would probably be quite concerned about loading up a road bike which is probably a more optimised design, and not set up for all of the additional loads being transmitted through parts of the bike where they are not intended. I would worry even more-so if you are being creative with mounting it in such a way that the loads may transmit twisting loads through the frame.
As for not enjoying the towing experience of the tagalong - you may be finding it unpleasant in the same way it is unpleasant towing a big trailer with too light a car - the trailer steers the car. There is a reason why towing laws (here anyway) state that the mass of the towing vehicle must be a certain percentage of the towed vehicle! I suspect the same should probably apply with taglongs to prevent the tagalong having too much influence over the ride on the tow-bike! They can be a bit deadly when the kids get all boisterous!
I have also helped a friend adapt her daughter to the same tagalong I used (a few other riders have used it in between) and the biggest thing is to train the kid that riding is a little bit serious. I sort of rode shotgun for the first little while to try to get the munchkin to sit evenly on the tagalong, keep both hands on the handlebars and look ahead. It makes a huge difference to the comfort of the tow-rider and stops them having that awful "tank-slapper" feeling when the kid suddenly whips around on their seat to point at something behind them that caught their attention.
Of course she was doing really well when I ruined the whole effect by singing the "there was an old woman who swallowed a fly" song to her as we rode along. When I got to the verse about swallowing a spider "who wriggled and giggled and jiggled inside her" the munchkin wriggled violently and nearly tossed her mother off the tow-bike!
That being said, mother and daughter (5yo) completed 4days of the Cycle Queensland without killing one another, so we did ok. CQ is a 9day tour of 500-700km depending on route.