Why would the Commander be incompetent? She's playing a game of extremely high stakes: she has to capture an enemy starship intact. The slightest misstep will result in Kirk activating his dreaded Corbomite Device and all is lost.
Trying to woo Spock against all odds really is a sensible way to proceed, then. Spock himself is a nice bonus (and possiby a consolation prize if all else fails), but the starship he now commands is the real prize. And if we learned anything from the previous Romulan episodes, it was that Romulans are a dishonorable race of backstabbers, hell-bent on short term gains and personal glory: the prize would have to be the Commander's personally, won through her own clever actions. Trying to bend Spock at least enough to make him slip something useful for the inevitable boarding and securing action (such as phrases she could record and replay!) isn't the only possible action - but it is an avenue she has no reason not to exploit.
Focusing on that at the expense of other things? Well, a tactical mistake, obviously. But what other designs would she have been working on? Probably ones where her personal attention was needed less, and where it would have counted for less. But again, she has to pussyfoot there, lest the ship blow to high heaven. Better work on what she has, full time.
While Starfleet no doubt specifically targeted something here (be it one of the ship formations verifiably equipped with the new device, or the Commander of known qualities), did the Romulans in turn target Kirk? He was a coveted prize, but unlike in "Deadly Years" or "Practical Joker", no Romulan ambush is required to explain the events. It's perfectly possible that the Commander was surprised and had to react and adapt, while Kirk and Spock were executing a carefully plotted overall plan, no doubt flexible but nevertheless specifically aimed to inconvenience the opponent. Starfleet really has the drop on the Romulans, dangling this bait and thus dictating the enemy moves. It would be pretty difficult for Kirk and Spock to actually fumble this one!
The rest is details. Why is Kirk the one to do the infiltrating? Out-universe: because he's the hero. In-universe? Might be because the Romulan sensors won't read an already duly logged-in (albeit deceased) alien as an intruder! Why are the shields down? Out-universe: so as not to slow down the action. In-universe: because they won't help against the corbomite blast anyway, but may hinder plans involving Romulan boarding action or cloak activation. Etc.
Timo Saloniemi