In all honesty 56cm sounds like a small frame size for someone who is 6' 1'' for some context I am 5' 8'' and generally find 56cm to be my preferred fit. All else being equal a 58 cm will likely give you more fitting options, as the frame stack. Most newer riders eventually complain of not being able to get the bars high enough, few complain they can't get them low enough (unless they are aiming for a particular aesthetic).

By swapping parts (e.g., stem, handle bars) and making adjusting it should be possible to get one frame size to fit like the other. The real difference between them is how extreme can you go. The 56 cm will be limited in how high (stack) you can get while the 58 cm will likely be limited in far you can shorten the reach. The taller (more stack) you get the more reach you also gain so I wouldn't be afraid of a larger frame.

It can often take many hours in the saddle before realizing one is a little too small or large (e.g., took me 1/2 year to realize one frame was too small causing me back issues). The best end goal I have found in terms of fit dimensions is being in balance, that is someone could quickly pull the handle bars out from under you and you wouldn't fall forward. The balance point will differ by some of your basic body dimensions and how aggressively you ride.

Smaller frames are generally considered for "performance" fits and tend to be pushed by many road shops these day. You combine a smaller frame with a longer stem to make up for the shorter reach. The idea is to get low and long. This is how the "pros" fit their bikes, and it is really intended for someone who is riding very hard effort all the time.

When you ride at a hard effort your core muscles are constantly engaged and the force of pedalling supports most of your upper body weigh. Often you will need to get your upper body as low as possible to provide sufficient resistance to the pedal force as well as improved aerodynamics. As a result you will have little weight on your hands and you can get away with a lot of bad ergonomics in terms of how far you are reaching.

In a race or a hard club ride I find this type of fit is great, and I typically find a smaller bike with a long stem (e.g., 54 cm + 110 or 120mm stem) works great. However, the moment I want to go on a slower ride with the kids or a friend I find this type of fit unbearable. When I was younger I could put up with it, but not so much anymore

Most do not want to ride at max effort everywhere. For an endurance fit (i.e., everyday riding), you want to be more upright and a larger frame  . You are generating less pedalling force so you will not be able to support your body pedal force if you are leaned to far forward. A larger frame makes this easier as the stack will be taller so you will have more to work with. If you are more upright your reach will often length so the slightly longer reach of a larger frame will also work well here.

In all honesty 56cm sounds like a small frame size for someone who is 6' 1'' for some context I am 5' 8'' and generally find 56cm to be my preferred fit. All else being equal a 58 cm will likely give you more fitting options, as the frame stack will be taller. Most newer riders eventually complain of not being able to get the bars high enough, few complain they can't get them low enough (unless they are aiming for a particular aesthetic). (Your reach often lengths with higher bars so you don't have to fear a larger frame.)

By swapping parts (e.g., stem, handle bars) and making adjusting it should be possible to get one frame size to fit like the other. The real difference between the two sizes is how extreme can you go. The 56 cm will be limited in how high (stack) you can get while the 58 cm will likely be limited in far you can shorten the reach. Also you can try even larger (e.g., 60 cm) as a reference point.

It can often take many hours in the saddle before realizing one is a little too small or large (e.g., took me 1/2 year to realize one frame was too small causing me back issues). The best end goal I have found is finding a fit that leaves you balance - that is someone could quickly make your handle bars disappear you wouldn't fall forward while pedalling. The balance point will differ by some of your basic body dimensions and how aggressively you ride on average.

Smaller frames are generally considered for "performance" fits and tend to be pushed by many road shops these day. You combine a smaller frame with a longer stem to make up for the shorter reach. The idea is to get low and long. This is how the "pros" fit their bikes, and it is really intended for someone who is riding at a very hard effort all the time.

When you ride at a hard effort your core muscles are constantly engaged and the force of pedalling supports most of your upper body weigh. Often you will need to get your upper body as low as possible to provide sufficient resistance to the pedal force as well as improving aerodynamics. As a result you will have little weight on your hands and you can get away with a lot of bad ergonomics in terms of how far you are reaching.

In a race or a hard club ride I find this type of fit is great, and I typically find a smaller bike with a long stem (e.g., 54 cm + 110 or 120mm stem) works great for me. However, the moment I want to go on a slower ride with the kids or a friend I find this type of fit unbearable. When I was younger I could suck it up, but not so much anymore.

Most do not want to ride at max effort everywhere. For an endurance fit (i.e., everyday riding), you want to be more upright and a larger frame. You are generating less pedalling force so you will not be able to support your upper body with your pedal force if you are leaned too far forward. A larger frame makes it easier to get your upper body in a more upright position (the stack will be taller so you have vertical height to work with) If you are more upright your reach will often length so the slightly longer reach of a larger frame will also work well here.