Someone asked if we're as skeptical of trailing participial phrases (those after the main clause) as leading ones (those before the main clause). Good question. The main answer is-- I'm skeptical of almost any modifier (participial or no) before the main clause. That doesn't mean I think they're all bad, because they're not-- often we need some essential bit of information (time or place, say) to get the full impact of the sentence. Additionally (just like that, in fact:), a transition (like "additionally," I mean) can serve to link this sentence with previous ones, smoothing the bump there to the new thought.
And sometimes what are called "phatic phrases" (relatively meaningless turns of speech) can add the necessary few beats for cadence, or make a polite deferral ("Needless to say"), or invite some ritualistic bonding ("As we all know").
That is, leading modifiers can have functions beyond actually modifying the subject (or the whole sentence, as many of them do). And of course, some of them legitimately modify the subject and fit well there, letting the reader know something needed to be known before she gets to the main clause.
However (another transition :), placement of modifiers ends up being more important at the beginning of the sentence. For reasons I'm not clear on, usually intro modifiers are "bound" in some way to the immediately following words. Well, you know, if you think about how readers figure out what a sentence is saying, they are accumulating meaning as they read, and in the beginning of the sentence, they need everything to go together in order to get the right meaning. So it is more important to have the modifiers modify the nearby words when those first six words are all there are so far. Bound modifiers are usually "restrictive," required for the meaning of the sentence because they restrict the modified word in some important way. ("The woman who had been waiting for him"-- the subject is not any woman, but the one who had been waiting for him.) So it's important if you have a modifier that restricts the modified word to keep those close together so they are read and understood together.
By the end of the main clause, the readers have a lot more context and perhaps don't need as much connection. (This is why, btw, dependent clauses at the -start- of a sentence are followed by a comma, but when they're at the end of a sentence, they're usually not preceded by a comma-- Because I failed algebra, I couldn't take calculus vs. I couldn't take calculus because I failed algebra.)
Modifiers after the main clause are more likely to be considered "free," and they aren't likely to annoy as much if they don't modify the nearest noun or if they're adverbial and modify the predicate instead of (as many adverbs do at the start) the whole sentence.
Now I'm kind of coming at this backwards. The issue is not actually whether a particular modifier goes at the beginning or end. The issue is that the start of the sentence is an important position, and that's why the reader is going to interpret what comes at the start as important. And if you start with something inessential, something that is "free" and unrestrictive and could go elsewhere in the sentence, you might be wasting that valuable real estate, that chance to set context or focus the reader's attention. Many of the modifiers that get edited out seem to have one reason for being at the start of the sentence, and that's not "context-setting" or "transition-making" or anything that helps create meaning for the reader. The reason often seems to be "varying the opening construction", and that is seldom sufficient reason to "bind" a modifier that might be better elsewhere.
What to do if you have four sentences beginning with "he"? First, determine if that's truly a problem here (sometimes it's good for rhythm or emphasis or alliteration or balance to start a group of sentences the same way). If it is-- if the repetition is annoying or distracting-- think about using what might be essential introductory material, like a time-marker (In 1816,) or a transition (However,) or a necessary bit of information about this subject (The president of Switzerland,) or what they used to call a "sentence modifier," a word (usually just a word-- not sure why) that is meant to affect the whole sentence (like "Frankly," or "Regardless,"). That is, find something introductory that adds to the meaning of the sentence, that starts the meaning, in some essential way. (Sometimes, yes, this can be a participial phrase-- just not all that often. Why? I think probably it's because a participle is supposed to be an action, and if it's an important action, it might be better in the main clause, and if it's not that important, it shouldn't be starting the sentence.)
Notice that most of these important introductory things are short-- a few words or less. (Intro dependent clauses, especially "If" clauses, tend to be longer, and often I end up making them independent clauses, or transferring the conditional-- if-- to the second position. But dependent clauses-- because they're clauses with their own subject and predicate-- can be longer even at the beginning... easier to understand and accumulate the meaning when there's a subject/verb.) Shorter intros will "hide" the repetitive opening ("he"), while offering some essential piece of information ("In 1816") without burying the lede (main clause) too deep.
Okay, that was as clear as mud. But do be thinking of the start of the sentence as a place where you want something important, even essential, to the understanding of what is to come. This usually just isn't a good place for a mere place-holder, though paradoxically it's often a good place for that "phatic" phrasing. So much of "polite speech" (which phatic speech usually is) is meant to beat around the bush, to blunt the sharp point of the sentence, and in that case, burying the lede (there I go with the mixed metaphors again) is actually sort of the point. Needless to say, in point of fact, as we all know :), in that case, go ahead, put soothing filler at the start of the sentence-- just don't dull the point so much the reader doesn't get it.