Do I need advance reservations on Eurostar?

Yes. You can probably buy them on the same day, but that will be expensive. Business premier tickets are fully refundable and changeable. Standard and Standard premier are changeable up to a week before departure for free, after that for a fee.

https://www.eurostar.com/us-en

Answers on their website.

Cheaper to buy in advance.

These are priced just like air tickets: better businessperson times of day, closer-in time trips, one-way trips are likely to be significantly more expensive. The cheaper tickets can be non-cancellable, non-changeable, non-refundable. You have to pay for freedom, nowadays. And these trains are reservation only, and they do sell out. How much that happens post-Covid, I can't say.

You need an advance ticket. Tickets are more expensive on Saturdays and at peak times during the day. Cheaper fares are very early or in the evening. I just looked for August and some services have sold out of the cheapest non refundable fares. The more flexibility you want, the more you will pay.

You need to be there at least an hour before departure to clear security.

Tickets for August are likely to be expensive already, so don't dally!