Presume you meant 200' long,not 200" - which might weight in favor of trenchless installation if in a deep burial zone.
OK - you had said HDPE, now poly - is this HDPE (which is tough as anything, assuming rigid rather than flex tubing), HDPE seamless flex tubing, or PE tubing (polyethylene, like drink bottles are made of) or more likely if rigid pipe, PVC ?
Personally, when you say under 2 driveways I assume you mean crossways to, not along the length) - not a big thing to patch driveway anyway unless you are a fanatic about appearance (because patch over a ditch will show)- but the cost of patching a driveway versus higher cost of trenchless installation - no comparison in most cases unless trenchless pipe installation is unusually cheap in your area. Plus, for 200' run in rocky soil, they would likely have to open up a drilling site halfway along to do a trenchless install anyway - the machines that can do 200' continuous runs are pretty expensive to operate. So, unless a lot cheaper - which probably means deep curial due to deep frost conditions, I would say trenched would be your safest bet - especially if your area has the occasional large rock. One major problem with trenchless is it tends to worm out voids under rocks that are large enough to stay in place rather than fall into the hole like sand and small gravel, so after you put the pipe in place the earth pressure, which tends to gradually close the hole up, can press a large rock into the pipe and puncture it. Of course, a lot more of a concern in shallow bedrock and mountain/canyon/steep creek areas where the rocks are sharp than in larger river valleys where they have been rounded.
If you have punctures in the pipe now, doing a trenchless install (unless heavy wall, expensive field-seamed rigid HDPE lengths) is likely to cause the same thing again in the future - I would generally prefer to trench and uses proper coarse sand or pea gravel backfill around the pipe (typically 4-6 inches all around) if all that you are damaging by doing so is asphalt driveway.
When you said sleeve, I assumed you meant sleeving your existing pipe - internally. Sounds like you mean a pulling sleeve - an outer plastic pipe, which the actual water pipe then goes inside. Certainly that would protect the water pipe somewhat from rock punctures, at least from small ones, certainly if it is a semi-rigid pipe rather than flexible, "soft" tubing.
I guess if you are comfortable with a pulling sleeve, then internal to that the actual waterline, I would give the contractor both options - that and trenched, at their discretion. Be sure to specify that backfilling where it crosses the drives has to be well compacted to driveway construction standards, with proper base material and asphalt paving to match current construction, all as part of their bid price. Ditto to repairing any damage to sidewalk - and replacement should include connection cost to city line. You will have to decide whether to require that be a new connection tap to the city main (costly at times), or if you will accept them connecting onto the existing tap - which may mean leaving a few feet of the old pipe connected to the main, depending on kind of tap - usually tap is at main (in street usually), with a tap line to the property line or sidewalk, where the shutoff valve box is - your connection point is usually at the shutoff valve.
Also - if sleeved, but sure the open end at the house is well sealed, so if you get a leak somewhere along the way you do not have an open passageway inside the sleeving to your garage. Usually use an expanding waterproof polymer foam for that,, several feet into the sleeve, though there are mechanical packings make for that too.
Connection to house - tunnelling under the house can have bad consequences - cracking foundation, settling slab, etc years later because the drilling removed material from under the house. I prefer to always come in alongside the house if possible, then a 90 and run the shortest distance possible under slab and foundation into house, then up into the ceiling joists to the old distribution point. And remember - if boring the line, they cannot hit an exact target unless usingthe old line as guidance, with a pilot probe on the bit (which not all can do) - within a few feet is considered normal, so coming up to a "clean" point on the foundation away fromthe front door and rhodie might be your best bet - maybe even other side of driveway on the garage ace (sounds like side away from the house), then an emergency shutoff valve (indoor if in freezing country) and new inside line (which could still be the larger service diameter rather than the probably 3/4" interior pipe size) inside the garage to tie into the existing piping.
Remember the connection between the outside line and the inside is a likely leak location so consider where that connection is made - maybe outside the garage rather than inside. That is the reason some jurisdictions requirecopper for the last x feet of run, plus it is generally more forgiving of foundation movements.
Remember also that copper and concrete do not mix - it needs to be sleeved through the concrete to avoid any close contact.