The bridgehead historically was populated by Russia's best remaining skilled fighting force (other than arguably Wagner Group): the VDV, airborne. They proved their tenacity and skill on the defensive over many months, beyond that of their abilities on the offensive (*cough*). There were 2-3 divisions of these in the bridgehead, making the bulk of the remaining VDV globally and presenting one of the highest-value targets for destruction in the entire theater of war. Removing Putin's elite from the field by one or other means would weaken his ability to establish strong defensive networks or train new troops for the remainder of the war. Also, it would give the Russian nationalists a stroke.

I do not believe the risk of losses to enemy action is per se a barrier to Ukraine in this war, since for the past 3 months they've run continual Russian-style offensive actions and frontal attacks throughout Kherson and elsewhere, losing hundreds of armored vehicles and probably several thousand men in the process. In the case of a pursuit action, in all but the most bungled execution the enemy will suffer an order of magnitude higher losses (as we actually saw during the September Kharkiv offensive). If UFOR is willing to assault RuFOR trenchworks from the front, taking a retreating force from behind seems less onerous. Giving away hardware positions is similarly not relevant when the enemy is leaving the area.
It's an interesting conundrum, I too think they would have been better off pushing an advantage harder. The dynamic between fighting the enemy versus fighting for territory are often at odds in war. The Ukrainian leadership must have chosen to not push too hard and save that equipment and manpower for later, perhaps the political value of this type of victory now is worth it.

Additionally, the Russians if pushed too hard would likely stand and fight, a trapped enemy is sometimes the most dangerous as they have nothing to lose. I'm sure the Ukrainian command has a decent respect for the Russian fighting potential when properly motivated and led.

But it is an absolute mystery why the Russians were not heavily bombed on the bridges and ferries out.
If the rumors of Russians being told to change into civilian clothing before fleeing are true then it may have been to avoid the propaganda value of what would look like shelling fleeing civilians that are pro-Russian. This would have the potential to be used for propaganda in the remaining occupied areas to show those that have collaborated or at least been neutral-ish to side with Russia openly or face the wrath of the Ukrainian 'nazis' cleansing areas of ethnic Russians post liberation.
Being a maneuver warfare guy I'm not too up to speed on effective information operations but the past eight months have certainly been eye opening.

Do you honestly think that Russian forces have a choice at this point?

Short of mass mobilization and attacking with >10 to 1 losses, I am not sure if there is any possibility of maintaining the campaign. Even if you do have some success, what then? With what are you going to hold? What are you going to hold?

Let's say that before spring, a massive campaign is launched and Russia suffers another hundred thousand to a quarter of a million losses. What would drive the economy? What would you do with the wounded? Could you exist without building primitive factories sending crude iron boxes to the front all across the territory? I don't think that anyone on the RF side has the appetite for the type of daily losses that Stalin would consider a rudimentary necessity. Could a society exist?
Those are absolutely valid and rational points. The question really has to be, is Putin and the rest of the Russian leadership still totally rational actors at this point? Given how poorly the war has gone and the likely prospect of a palace-coup if they are fought back to pre-Feb 24th borders or worse (for the Putin regime) the 2014 borders I can believe that they would sacrifice a million losses to hold what they have and force another 'frozen conflict' ending to be continued in a few years after licking their wounds. Perhaps the WW2 example of how much the USSR sacrificed to stop and then beat back the Nazis makes the Russian leadership too dismissive of casualties as that would make them weaker than the 'Man of Steel' and not fit to lead Russian resurgence.