Users subscribe and depending on subscription tier, will be able to book x number of beauty services anywhere (with partners) in a month. So I guess you could say its kinda like classpass but solely for beauty like spas, facials, mani/pedi.
1. Will the economics work out?
2. Do I also charge a flat fee or commission for beauty suppliers?
I expect that it will "work out" depending largely on whether you own the assets or if you're merely a referral service. Do your research, explore the models you're considering. As @Dane said, there are existing models out there like Massage Envy, Aveda, and others, including gyms, resorts, etc.
Likely there are some regular beauty services customers who use frequently enough that they would jive with a subscription model. What you have to validate is the marketing strategy and determine the value proposition.
Consider that almost all middle-men (agencies) are eventually squeezed out. That's why direct ownership often works better than referral businesses.
Look at Massage Envy - it is similar in a monthly fee for a tiered service. It is a franchise with hundreds of locations in the US. It cold be interesting to co-opt the private clubs and high-end hotels as early network participants.
No clue on the economics; it depends on your model and fees.
Do you mean providers or professionals when you say "suppliers"?
Do you have experience in beauty services?
Sure why not. But not this year. Maybe not next year. Everyone is in lock down. There is likely to be a second Covid bounce in fall.
You may want to offer a certification that each place follows certain hygiene protocols.
If you need help with developing such protocols, I can put you on to someone who can help you.
Are stylists interchangeable? I ask because if someone sees the same stylist multiple times, they'll cut you out unless it's cheaper through you.
This has the standard two-sided market problems.
The right economics will be the result of experiments.