The single best Pokemon to use as a hatcher is Heatran. As a legendary, it has impressive stats that allow it to deal a lot of damage and not take a lot in return, making it unlikely to be knocked out. With its amazing Fire/Steel typing, it resists 9 types, is immune to 1, and is weak to 3. This makes it defensive enough to take a hit, and with Flame Body, it can hatch Eggs in half the time it usually takes.
Heatran @ Leftovers
Ability: Flame Body
- Lava Plume / Flamethrower
- Earthquake / Earth Power
- Iron Head
- Crunch / Dragon Pulse
The unequivocal best Pokemon catcher in Sword & Shield is Gallade. With a high Attack stat, a boosting move, False Swipe, Rain Dance, and status moves, it can keep almost any Pokemon alive long enough to catch it. Its only two downfalls are not being able to block Self-Destruct/Explosion/Memento and not being able to touch Ghost types, but those cases are rare. An argument could be made for Pangoro, which has both Scrappy and False Swipe, but Gallade is typically better.
Gallade @ Muscle Band / Silk Scarf
Ability: Steadfast
- False Swipe
- Swords Dance
- Rain Dance
- Thunder Wave
If you want a single Pokemon that can both effectively catch others and hatch Eggs, your choices are more limited. Scyther or Scizor are good False Swipe alternatives, but neither have any way to make hatching Eggs go faster. For this reason, the best choice would probably be Magmortar. The other contender is Kantonian Moltres, but it's got a few things going against it:
- Moltres requires transferring or Dynamax Adventures
- Flame Body is the Hidden Ability, meaning you need an Ability Patch
Magmortar isn't the best choice either, seeing as it's a bit frail and can't earn False Swipe, but it's the best you can do to fulfill both the roles you want.
Magmortar @ Leftovers
Ability: Flame Body
- Sunny Day
- Taunt
- Facade / Will-O-Wisp
- Fire Punch
If you really want to use Moltres, it does work. As a legendary, it holds its own, though it can be a bit of an annoyance to get -- but if you're hypothetically willing to use Heatran like you said in the comments, that probably won't be an issue.
Moltres @ Damp Rock / Leftovers
Ability: Flame Body
- Rain Dance
- Mystical Fire / Facade
- Air Slash / Facade
- Roost