Thracia is pretty much the only Fire Emblem game to actually balance footies and mounties, partially due to the (based) dismounting system. As it turns out, movement is a really good stat, especially in a game where you can dump a shit ton of scrolls on one guy and cap all their other stats really easily, so you need to throw a bunch of limiting factors at your mounted units to make the player want to actually not deploy them every single chapter.
It's unfortunate that some of the mounted units can't use their primary weapon while dismounted, but by no means does this make them bad to use indoors, ruin indoor maps, or make the game not fully enjoyable. Take Finn like you said; yes he can't use lances indoors and he takes the movement penalty, but he's still got his good stats with a promo to boost them further, a leadership star, miracle, and his sword rank isn't even that bad to begin with. He's still actually better than plenty of unmounted units like Trude for indoor chapters despite the handicap. The only units who really get fucked by the dismounting system are units that are already shitters who have nothing going for them even when mounted, like Hicks, Alba, and Cain. They wouldn't even be good if they could use their primary weapon while unmounted; the mounting system is the least of their problems.
Yes, there aren't any ways to use lances indoors without Dalsin or Xavier. If this really needs to be changed I'd much rather it be done so by some recruitable soldiers rather than gimping the mounting system entirely. The mounting system is a deliberate and effective design choice used with a bunch of things like fatigue, low stat caps, and the general map design to encourage the player not to just use the same dozen or so units the whole game. It's one of the only games in the series that actually accomplishes this without just straight out forcing the player to use different units like Radiant Dawn, or to make enemy stats so damn high that you actually need to go off of the highest stats possible rather than mostly just movement, such as in PoR Maniac mode or Conquest/Awakening Lunatic. Basically, if you change the mounted system you're taking away part of what makes Thracia unique and messing with it's game design.
Side tangent but I've never gotten the hype for Three Houses mounting system. Wyvern Rider/Lord is still the best class in the game and being a mounted unit is still better than being an unmounted unit except for some niche cases. By no means do you really need every unit to be mounted to beat the game and it wouldn't be much fun to do that anyways, but the fact is that any magical combat unit (basically if you aren't a physic/warp bot) should be a Dark Flier, most physcial units are better off in Wyvern Rider or Pegasus Knight, if you really want a General it'd be better off to just make them a Great Knight eventually, etc. Why would you ever want to be a Swordmaster or Hero when you can just be Wyvern Lord who uses swords instead? Three Houses' dismounting system ironically doesn't do much aside from making mounted units better than they already typically were, and they didn't need any extra help. Although to be honest I've never gotten any major use out of it aside from occasionally dismounting horsies so they can walk through forests through all my Maddening playthroughs.