The Enterprise's interior definitely draws more from TOS than any other Starfleet ship seen in Disco does. They're even including all kinds of little details left out of the other ships, the bridge chairs are modeled after the TOS ones, the turbolifts have the handles, the briefing room has the triple-sided monitor. The Enterprise has all the familiar TOS sound effects as opposed to the other ships using sound effects from the 24th century shows. The uniforms are done in a definite TOS-style complete with the TOS style sleeve braids for the ranks as opposed to the other uniforms using 24th century pips etched onto the chest insignia.Tripping over themselves?
No I don't think so. Their version of the Enterprise looks more like the TMP version then the TOS version. The Enterprise ship interiors look like nothing like they did in TOS, and the uniforms also look nothing like they did in TOS.
As for Captain Pike, they do a really good job in keeping with the tone of the one episode we saw him in and I don't think previous Canon or continuity really hamperd their take on the character.
Overall I think they were respectful of Canon in continuity from TOS but no they weren't bending over backwards to do anything with respect to The cannoner continuity of TOS.
And this isn't even limited to the Starfleet side of things. The Klingons have had their look tamed down from being as drastically different as it was in the first season to something more in line with the look from the movies and the shows besides TOS, the bat'leths have been changed from the completely different look they had in the first season to something more resembling the traditional look, and most infamously the ship referred to as a D-7 in the first season is gone, replaced by the original design D-7.
The second season of Disco has made a very conscious and obvious decision to abandon where possible the first season redesign and go for something a hell of a lot more familiar.