I am afraid this will remain a nightmare to filter in one go for the reason that you mentioned:
since many emails contain the words "privacy policy" or "terms of service" in the footer.
But let's try to figure out what to do!
First step
What you could do is delete only messages that match exactly the groups of terms you never want to read as to not exclude important emails. To create one filter for all of them, you can use the field has words and enter something like the following:
"Privacy Policy Update" OR "GDPR" OR "General Data protection regulation" OR "Updates to our terms of use" OR "Updates to our privacy" OR "updating our privacy" OR "updated our privacy"
If you know more specifically how the words you are looking for appear, you can check for turns of phrases such as Updates to X's Privacy Policy and manage them with the keyword AROUND <number> where <number> is how many words can be around what you are searching:
"privacy policy" AROUND 3 update
This will look anywhere in the subject or in the content of the email and find the word update as well as updates near the match privacy policy. If you are interested in looking for the word update but not updates, you can specify an exact match with the + sign right in front of it: +update.
Next step
A next step could be to filter potentially important emails with more keywords but instead of deleting them, you can select a few options for your filter such as Skip the Inbox (Archive it), Mark as read, Apply the label and Never mark it as important. That way, you still have access to all those emails and you can review them separately and quickly at your convenience.
For a list of available search operators, you can refer to this Gmail help page — Search operators you can use with Gmail.
The extra mile
As a workaround, you can use a service such as s p a m g o u r m e t (sorry, had to edit as this is considered spam) to create as many email addresses as you require to login to most websites that you do not intend to use regularly and create a label Filtered mail or something similar. You can create addresses on the fly without ever needing to log into your s p a m g o u r m e t account and set how many emails can be sent to a specific address all at once ;). With very few extra steps, you can manage your email addresses, remove the ones you don't want anymore, reset how many emails can be sent, etc.
If you do not find this option useful, you can also use the + sign in your email address to further filter your incoming messages, which work with many websites. For example, you can set up your Facebook email to be [email protected]. That way, you know where your spam may be coming from, but it is always more effective to just use a unique address with the above service or even forward emails from another Gmail account so you never have to give away your main email. By doing this, you can then set up filters in your other accounts before they reach your main account and if they do get to your main account, you will then be applying all your main filters also!
I hope this will be useful to you. It is not always obvious how to use filters effectively since they have their limitations, but you can certainly make it harder for others to reach you!
