A fairly minor change

I think your take on the issues with the Circle of the Moon Druid is a fairly reasonable one. Having played a one through to level 14 I agree there is a strong incentive to ensure your Wild Shape is available for combat as it is one of the strongest combat features in the game.

I'm going to address your features seperately as I don't believe there is a strong reason for them to be tied together and players/GMs could easily use one or the other without consequences.

Native Forms

Given that you are granted a feature and taking nothing away this is inherently a power positive change, however I don't believe it is a bad one. A lot of the utility power of Wild Shape comes from being able to choose the best possible form for any given situation at a moments notice. By restricting to one (later two) forms you remove that ability and make this a fairly minor feature overall.

The benefit to roleplay and the "feel" of playing a Druid is far from minor. I and every Druid player I've played with would have loved this feature from a purely roleplay perspective. I believe you could even grant it earlier or raise the CR limit to 1 (but not both) without introducing a game breaking change. This feature doesn't really enable the Druid to do anything they couldn't do anyway but removes the need for taking a short rest immediately after, slowing down the game.

The potential for misuse is low due to the relatively weak forms and the potential for increased fun is high. That makes it a great change in my book and one I am likely to introduce in my own games.

Elder Druid

A Circle of the Moon Archdruid is brokenly powerful. It is by far the strongest capstone ability and can almost make them un-killable (par some instant kill effects). I believe the only reason it isn't a bigger issue is the rarity of level 20 play. For anyone that doesn't believe this watch Critical Role's Epic Level Battle Royal:

Where multiple other 20th level characters working together still fail to defeat the Archdruid.

Therefore it makes sense that you might like to address this power imbalance. Given that you are taking away one of the strongest abilities in the game and replacing it with a weaker one this is obviously a power negative change. I'm going to ignore the part of Elder Druid that matches Archdruid and focus only on the first paragraph.

Regaining Wild Shape at the start of combat is undoubtedly useful and would likely lead to players using their wild shape ability more as there is no risk of being weaker for a combat. The is similar to what you may see with the original feature.

Being limited to two uses during combat is an enormous downgrade, the loss of hp, combat flexibility and endurance from the change is difficult to quantify. But I would estimate it at far far higher than 5hp per round.

The healing from your feature is so minor as to be basically meaningless at this level of play. Expected damage would be in the range 50-100 points per round rather than per minute so the healing won't make a meaning different to combat. It is also unnecessary between combats as you regain wild shape at the start of each combat and can always shift into a new form instantly regaining maximum hp.

I suggest scrapping the healing feature and instead granting 1 or 2 additional uses of Wild Shape at this level. That should better mimic the feel and power of the original feature while removing the more broken aspects.

Conclusion

Overall these changes are relatively minor and unlikely to impact game balance in a meaningful way. Native Forms is a fun, very minor buff with few downsides. Elder Druid is a more significant de-buff but one far less likely to be relevant in most games.

I see no reason not to allow these at the table and encourage you to playtest and report back on if they resolve the issues you have encountered.