To preface, I’m going to answer solely from the perspective of a Wizard player as that is the only class which has access to all 3 of the spells you’ve described without invoking feats and the like.
Mage Hand versus Unseen Servant – Opportunity Costs
Any situation where the character needs to pick between two options and one of those options is a cantrip the character (especially a Wizard) has a limited number of cantrips that can be known. Conversely, 1st level spells do not have an effective limit, a Wizard can put as many in their spellbook as they can find.
In my experience playing as a variety of casters, the cantrips you select become very defining elements of your character as well (especially when they are non-damage dealing). For example, if my character knows the Mending cantrip, he’s never going to be seen with torn clothes if he can help it; if my character knows Message, he’s going to be using it whenever possible to covertly relay information between party members; and if my character knows Mage Hand, he’s going to often be moving things around with that while reading through a book or using it to test an area for traps or whatever other creative uses I can think of for creatively applying 10 pounds of force.
Conversely, if I’m a Wizard that doesn’t know Mage Hand and I wish to implement the features of it I can do so, but it will be more costly either in time because I need to ritually cast Unseen Servant or in spell slots so that I can expedite things down to an Action. In some ways, the Unseen Servant will be better and others not so much, however, what my Wizard pays in time or spell slots they will be repaid in opportunity with another cantrip selection.
Unseen Servant versus Find Familiar – Monetary Costs
The reasons you listed are more than enough to provide the pros and cons of each spell. They are what they are.
However, given you are playing LMoP, then you are in the 1st tier range of gameplay. Thus the cost of one or the other is measured less so in terms of which is better but rather in monetary casting costs. Find Familiar, as you noted, costs 10 gp to cast. From the backgrounds available in the PHB, a large number of them start with only 10 gp. Thus, if you wish to start the game with a familiar, you can do so but you’ll burn all of your starting money to do so and if the familiar dies then you’ll need to buy more material components to create another.
Obviously, this major limitation goes away fairly early in a character’s adventuring career. However, the journey between 1st and 2nd level is often the most perilous. As a player, you have to make a choice. Do you select Unseen Servant, which is free to cast ritually, or will you risk your starting resources to have the much more potent Find Familiar?