Correctly and incorrectly mounting a bicycle

You need to realize that sideway stresses do not equate to an angled bike. Instead, sideway stresses on wheels depend on the angle between the force that the road exerts on your tires and the plane of the tire.

Note that there is no "plane of the road" in this: The angle between the bike and the road is totally irrelevant. When you ride straight, your bike is vertical, and the force transmitted to your bike is vertical. When you lean into a corner, the force is not vertical anymore, it is directed towards the inner side to actually accelerate your body mass. And the direction of the force is again flush with the orientation of your bike. Otherwise, you would topple over.

The only way to put sideway stress on your wheels, is by moving your center of gravity out of the plane of your wheels. I.e., you need to go out of the saddle, and then slant your bike to one side. People pedaling hard do this all the time, rocking the bike from side to side, without moving their center of gravity along.

The extend to which out-of-saddle pedaling can stress the wheels is rather limited: You are still constrained by a) having both feet on the pedals, and b) the top tube between your legs (if you ride a diamond frame, that is).

The cowboy mount, however, does not have this restriction. On the contrary: You start with your center of gravity way out of plane. You put your entire weight only on a single pedal. If you start with your bike vertical, you are first accelerating your center of mass towards the bike (by pushing with your right foot), and rely on your tires to stop this motion (so that you can end up with the bike beneath you). And this acceleration is very much out of plane, easily putting more sideway stress on your wheels than when you are pedaling out of saddle.

That said, bikes should be able to handle the cowboy mount. At least if they are built for durability (* cough, cough *) rather than race performance.

You are more likely to fall off that way than if you do it correctly.

Put right leg over bike and into pedal clip on right. Raise right pedal. Use left leg to push off then lift body onto seat and insert left foot in pedal on left while starting to pedal with right foot then continue with left yada yada.

As postulated by others, this seems clearly exaggerated. While riding up a mountain hard (out of the saddle), a significant portion of the whole body weight is on each pedal all the time, and this can go on for hundreds of repetitions, depending on the endurance of the rider.

That said having only that technique for mounting your bike in your arsenal is still bad. One further point why the technique you mentioned (starting with one foot on the pedal, and then swinging up the other) is not optimal is that it is a dynamic motion. Even if it doesn't hurt the bike, it hurts your own technique. If you are in a rocky, steep nontrivial terrain, you do absolutely not want to start that way, but you want to find a technique that starts very static and controlled, possibly ending up in you doing a trail stand on your bike. It's all about control.

I would argue that any mounting technique that allows you to start riding the bike is correct.

I use primarily the technique you describe as I feel I'ts agile, stable and quick.

The argument of side load is bogus. If a bike broke during that maneuver It is unsuitable for riding. Also, it is the same as applying all of your weight on just one pedal. The angle would be the angle needed to keep the tire contact points in line with your body and bike's combined center of mass. In order to increase the angle, you'd need to perform a pretty weird mounting (or you are performing acrobatics rather than just riding).

However, there are a few situations where the cowboy mounting is less advisable or not suitable.

When riding a bike that has a loaded rear rack, the swing required may be too much to be comfortable, and depending on what you're carrying, the prospect of hitting your leg against it can be enough of a deterrent. (From experience I know it hurts a lot more on the downswing). Sometimes I've found myself carrying tall loads that would impede completely the cowboy mounting.

The other situation is when you are in a crowded space, like in a group ride or in a crowded street during a commute. It is very easy to loose track of whether you would hit somebody. You may check before, but for when you execute the movement, someone may have entered your movement path. You could also swing your leg too close for someone else's comfort. Finally, some loose matter from your shoe may fly into someones face, etc.

In those situations, for safety and for the sake of avoiding awkward situations, I mount by lifting one leg over the top tube.

Also, for starting on inclined, loose or irregular terrain, It's easier and sometimes the only way. When I need to start on a very steep uphill, on a mountain trail with loose rocks and roots I do leg over top tube, clip that foot (usually the right one) and lift the pedal near the top. At this point, I don't push on the ground with remaining foot in order to advance, I use the pedal instead. As soon as I start pushing on the pedal, lift the other foot and place it on the pedal. If done quick enough, I catch the pedal before the upstroke ends and is ready for the down stroke.

In conclusion, having more than one mounting technique is advisable and helpful, but ultimately anything that accomplishes the objective and does not hurt yourself nor others and does not damage your bike is correct.