With your experience with 14ers, you will have no trouble at all with the TMB. I have only summited Whitney one time, at the end of a multi-week hike from Yosemite on the John Muir Trail. There is nothing on the TMB that approaches the altitude or the rugged, rocky terrain of the High Sierra. The passes on the John Muir Trail can be 11,000 feet high. The highest you will face on that section of the TMB will be the Col du Bonhomme and Croix du Bonhomme at 78 00-8100 feet. But there could well still be snow there in June; just be prepared (maybe with hiking poles and micro spikes?).
On the other hand, early season hikers may be safer from the more recent hazards of the Mont Blanc and other Alp areas, resulting from melting ice.
https://www.chamonix.net/english/news/mont-blanc-glacier-collapse-global-warming
https://www.bbc.com/news/world-europe-49820542
Our hike in 2018 was interrupted at one point—-the day we left Courmayeur to hike the Val Ferret and on into Switzerland. The rain that fell on days 1 and 3 of our trek caused a massive mudslide that closed the trail. Our guides took us on a very nice alternate hike, with an ambitious 4400 feet of climbing, and then the driver and van met us and drove the group to our next overnight stop in Champex.