Eiffel tower question

I think there are 3 ways to climb ET - elevator to the top (gives
access to top and 2nd floor) elevator to the 2nd floor (access only to
the 2nd floor) climb stairs to the 2nd floor (access only to the 2nd
floor) There is no option to climb all the way to the top - even if it
was physically possible to do. Am I correct?

You are correct with the three options. You cannot climb stairs all the way to the top of the tower. The view from the second floor is just as good as going to the top in my opinion (but I don't like scary small elevators or heights).

Other threads on ricksteves mentioned that not all tickets go on sale.
Some said additional tickets go on sale 2 weeks before the date of
visit. Is that still true?

There are tickets released 14 days ahead of time, subject to availability. https://www.toureiffel.paris/en/news/visit/all-our-tips-buying-ticket-visit-eiffel-tower

Should I just give up, reach the ET on June 10th, stand in line and
see if I can get in - either to the 2nd floor or the top, whichever is
available?

June is a very busy time. I would get whatever tickets you can ahead of time and not count on waiting in line. If you do choose to do this, go very early or late in the day when it is less busy (and not on a weekend). Also, if you haven't created your account on the website, do that ahead of time so you're not trying to do that while looking for tickets. Good luck!

Get tickets for level 2. Last time we were there (in October) we had tickets to go to the top. When we got to the second floor, the line for those wishing to go to the top completely wound around level two. The elevator holds maybe 10 people? We would have waited another hour for our turn. And people in this line were shoving and trying to cut in line. Zero crowd control. Not a very pleasant experience. We settled for level two and regret wasting the money to buy level 3 access.

We found that tickets dropped at midnight and around 4:45 a.m. Paris time. Also, we got nowhere trying to buy tickets online until we changed the clock on our computer to Paris time.

I have been to the top (a very long time ago) and for my visit last June, I opted for the second floor and it was perfect. Great views and not a long wait. Note that I didn’t get into the ticket site in time for the tickets to the second floor, however, there were tickets that included the elevator and a glass of champagne and those were not sold out. I just looked on the Eiffel Tower ticket site and they also have those same type of tickets for the top floor, so you may want to check that out if the other top floor options are not available.

Thank you everyone for your replies. Appreciate it.

I am taking my granddaughter to Paris and she really wanted to spend her 15th birthday going to the summit. I did two trial runs checking to see if I could get on to the website and buy tickets when they became available at midnight. I found that it took about 20-25 minutes to get to the point of actually purchasing the tickets, but that it worked both times. As recommended, I set my laptop to Paris time. On the day I actually purchased the tickets, while I was waiting for my laptop to be able to connect to the date I wanted, I decided to try my iPad—which was not set to Paris time—and was able to complete the ticket purchase before my laptop had finished scrolling. I purchased tickets for Friday 6/19.

It is very hard to get tickets. I just went though it as well and got lucky and was able to get tickets after repeatedly trying for 35 minutes after the midnight open. However, when I tried to purchase additional tickets I would get to the payment screen and the website would say "payment refused" even though there is no issue on my end with my bank/card and I tried four different cards. Wonder if anybody else has experienced this.

If you can't get tickets through the website you can go on Viator or similar sites and book with a guide. It is three to four times the price but it will get you tickets. I am not sure exactly how it works with the guide companies and their ticket purchases but seems like they have some sort of special access/ability/allocation for tickets.

The expensive alternative is what we did:

Dinner at the Jules Verne Restaurant. they have a separate elevator to the restaurant which is just above the 2nd level. The views from our table were fantastic.
After dinner, the staff showed us their special entrance to the 2nd level and we spent as much time as we wanted on the second level. We were there in the evening and the light show started while we're there. It was a very unique perspective. But it got cold and even in our nice dress clothes and coats, we got chilled. So we went back to their gated door pressed the door bell button and they let us back into the restaurant. Back in the restaurant, we took their elevator back down to the ground.

No waiting. No tickets needed. But you have to book the restaurant in advance and the restaurant is usually fully booked.