Getting seated in restaurants for 7

If you all want to eat at the same time within the window when Parisian's usually dine then one table may be seated while the others have to wait... So either eating earlier, or making reservations may be the best options.

Definitely make reservations. Unless you intend to eat at a large brasserie for every meal, seating 7 could be complicated, if not impossible. Even if you just walk past and stick your head in at lunchtime to reserve for dinner.

Unless you have time to kill, it is always advisable to book a table, especially for dinner, especially at better restaurants (not necessarily more expensive, but better quality restaurants).

Paris is a town where booking a reservation for dinner, even if you do it that morning, is necessary to have a reliably good experience. Some places with need a few days, a couple weeks, even months for fancy Michelin pllaces. The neighborhood place we like best can be reserved that day -- but if you j ust show up they are often full.

With 7 it would be folly to not reserve unless you go to continuous servce place or a place that doesn't do reservations like Chartiers. (industrial food, wonderful room, cheap eats -- as good as what you will get most places for a lot more). To get a great meal you need a restaurant with a short menu and a reservation.

Brasseries definitely need a reservation. We've run into several that were already full a few days ahead of time.

Not necessarily cafés, but I reserve all restaurants/brasseries. For 7, you can almost guarantee a table together at brasseries if you call sufficiently early. Locations such as the La Rotonde, La Coupole, La Grande Brasserie, Le Café du Commerce, Auberge-Dab, Au Petit Riche, or Bofinger, among many others, can all accommodate large groups.

I am not sure what you mean by "split tables".

Thank you all for the important advice. I think for our family, peeking in a restaurant in the morning and if a thumbs up, make reservations for that evening hopefully will work best. Also learned a new word and will most likely use, brasserie. Can’t wait to go.

Quick heads up, at 7 people you're above the online booking cap most French systems use (TheFork, Zenchef, restaurant sites), which usually stops at 6. So you'll hit "please call us" a lot.
A few things that actually work. Email instead of calling, plenty of places list an address and it skips the phone entirely. Write it in French, DeepL does fine, something like:
Bonjour, je souhaiterais réserver une table pour 7 personnes le [date] à [heure]. Nom : [votre nom], téléphone : [votre numéro avec l'indicatif]. Merci de me confirmer la disponibilité. Cordialement, [votre nom]
Give a number they can actually reach you on, and expect a day or two for a reply. Or just ask your hotel front desk to call, totally normal request in Paris, costs nothing. And honestly, for a group your size, aim at the big brasseries, Bofinger, Le Grand Colbert, Vaudeville, La Rotonde. They seat 7 without blinking, used to tourists, English spoken.
Book the important dinners 2 to 3 weeks out since October fills up, take a 7 or 7:30 slot if you can, easier to place before the 8:30 rush, and reconfirm the day before since a table that size sometimes gets released if they can't reach you. For any night you don't want to plan at all, the bouillons (Chartier, Bouillon Pigalle, Bouillon République) don't take reservations and happily seat groups, you just queue.

peeking in a restaurant in the morning and if a thumbs up, make reservations for that evening hopefully will work bes

As others have advised, I really would not operate this way for a group of 7. I would reserve a week to two weeks ahead of time.