You typed out a whole lot for something I already knew and acknowledged.
"But it isn't, because if this character does not make the switch, having the proficiency means that shields are usable in an emergency
with no penalty..."
It is impossible to use a shield with "no penalty" because it weighs 6 lbs and takes an action to don. There is no way around those and proficiency does not cancel those "penalties".
And you can absolutely use it without getting disadvantage on ability checks, attacks or spells just by using your action for something other than an ability check, attack or spell
My non-shield proficient character can use a sheild with "no penalty" other than the weight and action cost. You can dodge all day long with a shield and no "penalty" (other than weight and action economy), you can use help all day long with no penalty. When that hypothetical emergency comes up and you really need to don a shield, you can still do all kinds of things without penatly.
Now you can't attack without penalty, but two things here - first an attack must not be that important if you are using an action to don a shield. Second if you are not a shield user then your attacks will usually be less effective while carrying a shield. So not only are you losing an attack (one penalty) your attacks are less effective (more penalites).
Disadvantage can kill. As can that 10% if the monster does enough damage. Sometimes the player will want to use the shield without penalty in an emergency.
If you are wasting a turn donning your shield an attack is not that important. Further the haracter we are talking about uses two hands in fighting meaning his attacks are less
Further the difference in damage is not relevant except over a large number of attacks. Damage relative to chance to hit is a MEAN calculation, for it to matter you need to be targeted a lot. The +2 does not cut the damage you take at all, not one bit. Every hit you take does the same damage whether your AC is 5 or 35. The higher AC only cuts the number of times you are hit and for that to matter with a 10% reduction you need to be targeted a lot.
I am not suggesting a shield is not useful in protecting you, I am saying it is not useful in protecting you in an "emergency" situation like you are describing. Certainly not as effective as say masons tool proficiency or carpenter tool proficiency would generally be in an "emergency" situation.
They will typically come into play extremely rarely. typically far more rare than a situation where you might want to use a shield.
Those checks will happen way, way more often than a character who does not use a shield, using a shield, especially since such characters are not usually even carrying a shield.
I have used skill checks to find out information from paintings or glyphs numerous times in numerous campaigns. Plenty of times there are murals you need to decipher and paitners tool proficiency would give you advantage on that. Most of the published WOTC campaigns are going to have this in the campaign somewhere. DIA, TOD, TOA all have this because I remember specific times from those campaigns, most of them have it multiple times.
I only had a character who did not use a shield use a shield once and that was not an "emergency situation". It was a duel, she was a dex fighter who usually went around in leather and a bow, someone in our party had to duel a champion to save the town. She volunteered and put on chainmail and a shield.
That is one time, in one fight, in one campaign out of dozens of campaigns and hundreds (thousands) of sessions I have played. FWIW she also lost the fight so the shield did not save the day.
I will go a step further - I can say confidently100% that I have been in at least 2 campaigns that went 10+ levels where no one in the entire party used a shiled in the entire campaign. I can not say with confidence that I have ever played a campaign that went that long where no one in the party tried to decipher a Glyph or painting or other clue. I may have, but I can't say confidently that I have.
And please keep in mind we are talking about one of the least useful tools there are and it is still going to be more useful.
You think that just because Tasha's allows this trade that armor is = to a tool?
If it wasn't no one would make that trade it would be a waste of words. FWIW I have used TCE to trade armor for tool proficiencies, usually alchemists tools or thieves tools and every time it was an upgrade on that character.
It depends entirely on the build and on this particular build (someone who does not use a shield) painters supplies would be a small step up. Something like carpenters tools, cartographers tools, thieves tools or alchemists Tools would be a major improvement.
There's no way an armor proficiency is equal to a tool. Usefulness in 6-8 encounters per adventuring day is far, FAR superior to being rarely useful.
Armor proficiency will never be useful for 6-8 encounters per day on average unless you are only proficient in a single type of armor.
If I have proficiency in heavy armor, medium armor and light armor I need to have at least 18 fights in a day for all those proficiencies to be useful in 6 encounters a day and I need to change armor between fights to make that happen and presumably carry all that armor around with me all day. Finally you would be hard pressed to build a character that would benefit from switching between all three armor types (you can easily build one that benefits from two).
If you are wearing plate all day your proficiency in medium and light armors is useless that particular day and anny day that you don't carry those other armors and change into them at some point.
The same is true for shield proficiency. If you never use a shield then shield proficiency is useless.