The reality though is that this is modern state of SaaS-based web development. With Wordpress it’s the plugin store, Wix, etc. have similar. Webflow’s infrastructure works the same way but is only beginning to offer more seamless integration options in the form of apps.
In general emailed form responses only work well for small businesses, for others you may want to try a lead management system like Hubspot, or a Make automation that pushes your form submissions into Airtable.
If you’re managing the lead delivery yourself, one approach is to setup a Basin account just for your agency, and then create a Basin Project for each of your clients. Using a single Basin account makes it much more affordable, and you can easily build that $10 into your hosting fees overall, split across many clients.