You are not necessarily right:
"The Luftwaffe test pilot Lothar Sieber (7 April 1922 – 1 March 1945) may have inadvertently become the first man to break the sound barrier on 1 March 1945. This occurred while he was piloting a Bachem Ba 349 "Natter" for the first manned vertical takeoff of a rocket in history. In 55 seconds, he traveled a total of 14 km (8.7 miles)."
"The Walter liquid-fueled rocket motor built up to full thrust and Sieber pushed the button to ignite the four solid boosters. Initially, it rose vertically. at an altitude of about 100 to 150 m (330 to 490 ft), the Natter suddenly pitched up into an inverted curve at about 30° to the vertical. At about 500 m (1,600 ft) the cockpit canopy was seen to fly off. The Natter continued to climb at high speed at an angle of 15° from the horizontal and disappeared into the clouds. The Walter motor stalled about 15 seconds after take-off. It is estimated the Natter reached 1,500 m (4,900 ft), at which point it nose-dived and hit the ground with great force about 32 seconds later, some kilometres from the launch site. Unknown at the time, one of the Schmidding boosters failed to jettison and its remains were dug up at the crash site in 1998."
The Walter engine has 2000kg and the four Schmidding boosters 4x500kg thrust, but the Natter weighs only 2232kg. This is 1,792 thrust/weight ratio against the Bell XS-1's 0,495 value. OK, I know the Natter has worse aerodinamic design and was a subsonic plane, but not impossible.
Here is another article about this deadly flight:
Historical Footnote: On March 1st 1945, did Lothar Sieber become the first person to break the sound barrier?Yes, it’s true, Chuck Yaeger probably wasn’t the first person to break the sound barrier in manned flight. On a cloudy day in 1945, accomplished test pilot Lothar Sieber flew a Natter …united-cats.com
(It wasn't the only event, because Mano Ziegler reported his friend Heini Dittmar's famous flight on 6 July 1944 in the Me 163V-18 (VA+SP) at a speed of 1,130km/h. And Hans Guido Mutke claimed to have broke the sound barrier with his Me 262 on 9 April 1945 also. Without knowledge of the exact flight altitude, among other things, have not been proven.)
I'm quite familiar with the Natter and the Me 163.The Me 163 could reach 621 mph. Compression was well understood. The Me 262 had leading-edge slats to improve high speed handling. The sweep back also delayed compressibility. A captured film shows the Me 163 climbing at an 80 degree angle. The Natter was not just tested once. As a rocket propelled point interceptor, it reached altitude and fired its rockets or cannons, and on the way down, a lever was pulled as the aircraft fell to earth, releasing a portion and leaving the pilot free to parachute to the ground. I have seen a photo of a field filled with completed Natters. US soldiers/intelligence were very interested in this aircraft. In fact, it received the first FE or Foreign Equipment number, FE-1/T-2 1.