First. Let me explain some bits about the Play Store. You are using the Google Play store, so of course, it stores the list of installed apps on your Google account. That’s fairly obvious. However, the relation with the apps generally end there. Some apps might have Google account integration (Google Games for example, or just as a login method) but not more than that. There’s also potential Google ads and Crashalytics, but it ain’t “directly” connected to your Google account (It link your device through an unique ID, which Google could correlate with your logged Google account on this device)
You can download their app separately of the Google Play Store. Throught their website, source-code release or F-Droid (Depending of their project. Proton isn’t consistency with those)
ProtonMail don’t care about your Google account, outside of importing your emails into your new Protonnail account if you choose to do so.
Also, there is still a privacy gain, from using an email address from Protonmail, but only when you use it. It is just a better and more private e-mail provider. If you keep using a Gmail and Outlook email, than you will still issues with those emails addresses be used and read, for ads and analytics.
Do note you can use your protonmail address on your Google and Microsoft account. Those services just tend to automatically create an email account on their respectice services by default (Gmail & Outlook) but they aren’t required.