It's true that we didn't press the opposition in the manner that Birmingham pressed us on Friday, but McBurnie was certainly tasked with pressing Lee Evans (which is novel for McBurnie, who was conspicuously poor at this aspect of play last season) - as this article explains: https://swansea.vitalfootball.co.uk/analysis-how-graham-potter-won-the-tactical-battle-to-get-off-to-a-winning-start-at-sheffield-united/
Having watched the match again (it didn't get any better for viewing again) it has to be said that Monk won the tactical battle hands down. The tactics were pretty simple - allow us to have the ball in our defensive third, but as soon as we got within 20 yards of the half-way line they pressed us hard and collectively. Almost all their chances came from them taking the ball off players dawdling on the ball (Carroll being the main culprit) or intercepting over-long passes.
It disappointed me that we conspicuously failed to build from the back using short sharp passing triangles - with the bumpy pitch with long grass making long passes problematical. Olsson and Roberts (especially the former) were clearly instructed to hoof the ball long - with no positive end-result, although it did mean that Birmingham were regaining possession in deep-lying areas where they didn't pose a threat.
We had more of the second half through the simple adjustment of hoofing the ball long - relying on the lack of creativity in the Birmingham side to build anything meaningful from the back. It was a pragmatic alteration by Potter, and one that got us the point we didn't deserve - but it was still a tactical defeat, in that we abandoned our principles for the sake of short-term expediency. I'm not blaming Potter for that - in fact given the lack of a half-decent DM he had little choice. But it doesn't augur well for the future, unless we add a DM at the very least. One who could also cover at centre back would be ideal, so Besic and Ampadu both fit the bill. I suspect Chelsea will hang on to the latter to play in the Europa League group games and Carabao Cup.
Potter's got his hands tied for sure and that may go a long way towards explaining the pragmatic tactical choices he's making. it's good that he's flexible and not too dogmatic, but I hope he doesn't overdo the pragmatic stuff. Re-establishing our short-passing identity is going to take time and plenty of work on the training ground. I guess we have to have faith in Potter that any short-term pragmatic expediency of the kind we witnessed on Friday night is just that.