Family Eurorail pass?

I only count 8 days. Are you stopping somewhere enroute between 2 of the cities you mentioned?

Can't help with whether a Eurail pass would be better than point to point individual tickets. The only way to know is to check the prices on each of the train lines and add them up. Some of these trips will require an additional (per person) seat reservation fee if you have a Eurail pass. Have you added these into your calculations? And reservations should be made well in advance, as some train lines allocate only a small fixed number of seats per train for these passes.

Rail pass costs for your family can be found here:
https://www.eurail.com/en/eurail-passes/global-pass

Note that train journeys in the UK won't be covered, that some countries (like France) require seat reservations and restrict the number of seats available for rail pass users (which could make using your rail pass impossible in some places.) You should consider including some flights, and also consider traveling as Europeans do - using train TICKETS rather than passes - if you want a more economical, more streamlined, and more doable trip. I would suggest the following itinerary adjustments and sources for ticket information...

• Fly London - Venice (check google flights)

• Train from Venice to Salzburg

• Train to Munich (this journey can be done by local ticket bought at the station for €49/5 persons.)

• Train (+ DB bus) to Bern (probably about €40 each using a saver fare - see German Railways itinerary page)

• Train to A'dam

• Train to Bruges

• Train to Paris

• Eurostar train to London - https://www.seat61.com/London-to-Paris-by-train.htm

(For the 4 train trips in bold with no suggested source, get advice on your options from the "seat 61" site below.)
https://www.seat61.com/Europe-train-tickets.htm

"We spend approx 4 days in the following locations..."

If you have 4 NIGHTS in each place that translates into 3 DAYS when you figure in travel time - many of those trips are long ones. Also, some places probably require more time (London, Paris for example) than the others.

The Global Pass is valid on the EuroStar.

That is quite the adventure and I suspect by the end you will be an expert at navigating train stations and schedules.
Normally I'd scream to stay away from rail passes but with your itinerary (several multi-leg, single day trips) and group, it may just be worthwhile. Especially if it includes the Eurostar and considering you are booking late and the best fares are probably already gone. I prefer first class anyway, so I don't consider that in the cost analysis. I'd be locking down any required reservations ASAP.

Are you open to some of the itinerary changes that Russ suggested (or in the reverse direction) or do you have hotels and other plans already booked?

There is not currently a 9-day flexi version of the Eurail Global Pass. The average price per person is currently about $375 for 5 days within a month, $460 for 7 days, or $565 for 10 travel days, all in 2nd class.

Your kids are too old for free travel, but would save a bit with the youth rate, while the two adults qualify for a traveling together discount. In our shopping cart, be sure to check the box to apply that rate. For a limited time, see additional discount codes in our shopping cart that you can apply before check out.

The more expensive seat reservations are for the Eurostar and for a direct Switzerland-Paris train (TGV Lyria) - see rates on the link above. To and from Italy, the reservations are about $12 per leg. If you can commit to some of these dates and times, add them to your booking before check out.

And save time and money by flying home from Paris or Amsterdam instead of wasting a day going back to London.