Wiring – Dual oven with 6 wires

"The bridge between the screws was not broken" Are you 100% sure of that? Because the common use for black and red wires and a two-pole breaker in a kitchen outlet circuit is for a common neutral circuit (multiwire branch circuit), and shorting the two phases together with the new outlet would produce exactly the symptoms you have seen. So I'm going to suggest that the two sides of the old outlet are not actually connected, and if you don't have a meter you can check that with, I'll provide you an alternate test that does not involve looking at the connection, which you've already done and come to (I suspect) the wrong conclusion about:

Cap the red (or black) wire and connect only the black (or red) wire to the old outlet. Plug something into one receptacle, and see if has power. Then plug it into the other side. Repeat with the other wire (only) connected to the other side (only.) If I'm right, in each case, only half the outlet will have power, and which half will change with which screw you connect to.

It appears that the device you have probably only has one hot screw, so it cannot be connected the same way as the old outlet with two separate hots; so leave one wire capped off, or install it somewhere other than your kitchen (the reason for using this type of circuit in the kitchen is to provide more circuits to the countertop outlets - this device defeats that if it cannot have two separate hot feeds.)

Critical issues here :

• Side tab cut, not broken off. That is not safe as the 240V between them could easily arc, illegal per NEC 110.3 as the receptacle's instructions say to break off the tab entirely, not cut it. This defeats the safety testing done by UL since UL did not test it that way.
• Backstabs are not compatible with 12AWG wire, the wire size now required in kitchens.
• Bending the wires onto the screws is very shabbily done (poor conductivity, and NEC 110.12).
• Backstabs and screws both in use (I believe that's not allowed).
• Two separate circuits feeding the same yoke, but they are not handle-tied for common maintenance shut-off. (Alternatively, a 2-pole breaker can be used to accomplish the handle-tie, but never a double-stuff/duplex/2-in-1-space type.)
• Neutral must be pigtailed - correct. This is so a receptacle can be removed without severing neutral for the other circuit.

The fix is what you did: pigtail the hots and use screws only. Do that on the other receptacles... or, switch to the $3 Leviton screw-to-clamp type, which support 2 wires under each screw. And break the tabs off properly, they have break lines, a few flexes should do it.

When you are able, put the 2 circuits on a 2-pole breaker or handle-tie them.