You received many good recommendations from the other respondents.
The data which you provide indicates you need a classroom for 25 people four days a week for one week in each city. You visit 50 cities per year. You also want to achieve maximum cost efficiencies.
You suggested that working with one hotel chain might offer you the best value. It might, or it might not. The question you need to address is what is most important for you.
If you are marketing very high end products and want the best location in each of your 50 cities, then staying with one chain is probably not the best idea. After all, in one city Chain A may have the best facility. In another city, it may be chain B, and in a third city chain C.
You mention cost effectiveness. If you are looking for the least expensive venues, you might be better off staying away from hotel chains completely. You might be able to rent space in a religious institution, for example, during the week for a fraction of what a hotel might charge. Additionally, you would probably save a lot of money on Food & Beverage (F&B), and would probably have free parking. Likewise, you might rent a summer camp (or a school classroom) in the off season for very little money and have a really nice venue.
What the single chain would offer you is ease of making arrangements... but there would be tradeoffs. They may also offer you very deep discounts... you would need to find out how much they would be willing to discount space as well as F & B. F & B can easily cost many times the room rental fee. Indeed, I once changed my choice of meeting location because of the cost for bottled water at the facility I was originally planning to use (that was not negotiable from the facility's point of view).
You may have an even easier time working with a quality independent meeting planner (or a talented in-house planner). They will be able to find you the best facility for your needs in every city (not every major chain is in every city), optimizing for quality of facility, convenience, cost or whatever you need to optimize for. They will also be able to help you (if you have scheduling flexibility) schedule your cities to optimize on low rates... and avoid peak times as much as possible.
Please do feel free to contact me if you would like to discuss. I have lots of experience with professional meetings as a speaker and consultant, manager, and board of directors member. Ron
Ronald G. Shapiro, PhDEducation by Entertainment