Branding, as with a lot of graphic design, is seriously misunderstood by those who don't really know it well. You either get it, or you don't but most people think they get it no matter what.
Sure, it's less of a deal for a small tourist RR or museum than a large company that operates around the world, but the branding says who you are, is a clear and instant ID for said brand, and when you go around changing it each time someone wants to put their stamp on things (and yeah, that does sometimes happen in the corporate world in those cases where a new honcho shows up looking to make a 'change' which they never get is going to be changed to something else by the next person), you lose something.
A logo isn't something you change every time you want to put the museum's name on something new. That makes as much sense as changing the group's name each time something new is done.
The problem in this context is that each time something is changed, it's probably with volunteer effort. Someone comes along and offers to have new T-shirts or other swag made. And maybe it's done with a donation or volunteer effort by a company that makes such things. So, they think, "Hey, I'm doing this for free, so I'll put our own spin on things," or the person in your group thinks that. Rarely will anyone then say, "Uh, wait a sec, our logo doesn't look like that..." because we all know what'll happen:
The volunteer will say, "Okay, fine, you don't want my 'input'? Then do it yourself!"
Now keep in mind, in a lot of cases, non-profit logos are usually designed either by a volunteer nobody wants to tick off or in many cases, their kids (really, it happens a lot even in the corporate world, I used to re-design children's stuff for companies all the time into useable logos when I did this for money after I got my commercial art degree). Many logos are poorly designed or sometimes blatant rip-offs of other work.
In those cases, you might be doing the group a favor of permanently changing to something else.
But you should never go with the mindset of, "this logo is almost 2 years old. I'm bored with it, let's just do something else!"

You are confusing an official company logo with merchandise design.
They are two different things.

If you care more about your one logo than designing merchandise that actually sells you've really missed some important sessions of business 101.

If you look at Disney Shopping's Disney World merchandise collection you'll see a variety of fonts, logos, and designs.

https://www.shopdisney.com/clothing/men/t-shirts-tops/?prefn1=disneyParksMore&prefv1=Walt%20Disney%20World&srule=default-sorting-rule&start=48&sz=24

Any organization should care quite a bit about their logo design and that is an important part of merchandise design. Merchandise should reinforce your organization's identity in the eyes of the public. We should make every effort to assure that the official logo is used in a way that is consistent with the approved design, fonts and colors so that everything from the sign out front to the brochure, website and coffee mugs all remind the public about your organization.

Don't use the Disney example. They are so big and well established that they can get away with having different fonts, logos and designs without it impacting the public perception of the brand. That will not fly with a small entity with limited reach.

The importance of a unified corporate image as reflected by the logo and brand cannot be overemphasized. Your name should be easy to remember, and if it is too long, think about a shorter brand name to emphasize instead. The logo itself should be simple, crisp, look equally good in B&W and color and remind people of your organization. Letterheads should match envelopes should match the website should match the brochures should match the signage. Uniformity promotes recognition.

The average person out there, your target audience, is not going to pay much attention to details and nuances, so the easier that you can make it for them to notice you the better. Using different logos confuses them and that can result in lost visitors or sales.

The lack of unified branding is something that I fight on a regular basis with my contract website and marketing work through my publishing company. My clients are primarily non-profits, and with only a few exceptions, most had multiple logos that seem to have appeared whenever someone new took over the design of any given material (be it brochures, the website, the merchandise, etc.). Getting a single logo/image/brand together for my client is usually my first course of action, and in several cases it's been a sticking point because I will not proceed with my design work until the basis for it is established. A lot of non-profits don't even view it as a necessity, so they're very confused by my insistence that that really needs to be the first step if you want me to do the things you hired me to do.

A great example of what a unified design can do was the New Haven Railroad Historical & Technical Association. They had multiple logos designed over about 20+ years when I started their website project last year. Unlike most groups, however, they fully embraced my speech about branding and took it to heart. They've got a brand-new logo that I designed and have used it on EVERYTHING to create consistency across all of their products and marketing material, and it seems to be working for them quite well.

This brings up another interesting point - a lot of groups use logos that are significantly based on the logos of original railroads that may still be trademarked or otherwise protected by their heirs. The Conrail Historical Society actually has a licensing agreement in place with Conrail (Shared Assets Operations) for the use of the "can opener" logo, since we use it on much of our merchandise and in our own CRHS logo. Some groups, like the Nickel Plate Road Historical & Technical Society, actually own the rights to their railroad's logo and typeface. But a lot of smaller groups don't have these protections in place and have logos that probably cannot be trademarked on their own since they would violate the trademark of the original railroad. The New Haven group is a good example of this - their new logo is designed based on the iconic New Haven script logo, and it may still be protected, in which case they would need a licensing agreement rather than a trademark.

I disagree with this.

The #1 goal of merchandise design should be designing merchandise that your customer wants to buy. That will do more for the bottom line of any business.