You're basically describing the Space Shuttle.
The Space Shuttle wasn't even a good solution when it was designed. It had precisely one goal - to look like a plane for the image of the Air Force. As far as engineering goes, the Big Dumb Booster was already well proven, and is what every other solution to space has used. But in order to get Air Force funding, it had to look like a plane. The justification at the time was reuseability - but even at the time, the level of reuse and cost of refurbishment made this questionable, and it was never a problem which actually could be solved because the design simply didn't make it possible.
Your first problem is reentry. Wings are really bad for that, because they're exactly the kind of shape you don't want, especially around the wing root. The leading edge of the wing is also a very vulnerable point. Wing-shaped heat shields basically aren't a good thing, and Columbia demonstrated what happens when you get unlucky. How many Columbias can your winged spaceship program tolerate?
Your second problem is landing it. The Space Shuttle was actually a glider - and as every glider pilot can tell you, landing is stressful. If the pilot of a powered aircraft gets approach wrong, they can push the engines up and go round again. If a glider pilot gets approach wrong, something is going to get broken, and that something is often the occupant(s). There are no second chances. So instead of a highly-automated system of engines and sophisticated control systems, you've replaced this with a pilot who can and will screw up, with no backups or any way to recover the situation. Sweet.
Ah, you'll say, but gliders don't crash that often. Firstly, you're probably not going to hear about it on the news unless someone dies. And secondly, gliders have a glide ratio of around 30:1, are ridiculously manouevrable, and land at about 40mph (fast but not too much) or slower if they can land into a headwind. The Shuttle had a glide ratio of 4.5:1, was notoriously hard to control (because a wing which forms even a partly-acceptable heatshield is not a good wing to fly), and landed at over 200mph. For reference, a glide ratio of 4.5:1 is substantially worse than any hang-glider - in fact it's about the same glide ratio as a skydiving parachute - and that landing speed is faster than an F-16 touches down. This is not a happy place to be, and it's a testament to the insane skills of the pilots (and a healthy dose of luck!) that none of them were lost on landing.
Thirdly, you need somewhere to actually land the damn thing. The big bonus of landing vertically is that you only need a flat patch of ground the size of your landing pads. The Shuttle needed a 3 mile runway to land on. That's 3 miles of perfectly flat ground, with the Shuttle initially rolling at 200mph when it touches down. Good luck with that on Mars.
And fourthly, you need an atmosphere. Earth has a thick enough atmosphere that wings work fairly well. Mars's atmosphere is a whole lot thinner, and designs for aircraft on Mars (there's a helicopter due to land next week!) need to pay serious attention to this. Glide ratios and speeds would be correspondingly worse - as if they weren't already bad enough for the Shuttle on Earth. And the Moon of course has no atmosphere at all.
If you really, really want a wing, then you can go back to Rogallo's work instead. Re-entry uses a normal heatshield, atmospheric braking with drogue chutes gets the speed down, and then a flexible wing is deployed. Since the invention of the ram-air parafoil, it's probably more practical to use that instead though. It may not look as cool, but you can still fly it as normal - it is perfectly practical, fairly robust, and easy to control. It even lands slowly. On the downside though, you have an extra thing to deploy, and any skydiver can tell you that chute deployments do fail, even before you add the extra ways that something can fail when you need to deploy it mechanically from a space craft. SpaceX did consider this, but their assessment was that firing the rockets (which after all are known to work, because they got you off the ground in the first place) is more robust than adding something else which can go wrong.
TLDR: It's not because of the weight of the landing gear at all!