Web wallet that support alt coins must include bitcoin cash and bitcoin gold.

[sunsilk]

~snip~

Coinomi is one of the most secure web wallets on the crypto market.

I've used it and I don't have problems with them but there might be people who will argue with you on this for telling it the "most secure" as it's a closed-source wallet.

That's right, coinomi crypto wallet app is not a web wallet but I trust this wallet more than the web wallet. Dont trust web wallet in storing big amount your bitcoin, because you are not holding your key and which means you also not holding your bitcoin private key.

I also dont have any problem of coinomi, so far so good and I can able to accept variety of cryptocurrencies through this wallet app.

For list of all wallets, this is the accurate answer.
https://www.cryptowisser.com/wallets/

There's just inaccuracy with what he said about being the "most secure" yet it's not an open-source.

For example sometime back, coinomi was discovered to have a vulnerability where the wallet would send your plain text seed phrase to google's remote spell checker API - https://www.reddit.com/r/Bitcoin/comments/av987o/security_vulnerability_coinomi_wallet_sends_your

I still remember this issue and that's why even if I use them sometimes, I don't usually store any crypto there and I rarely use them as a cold storage although they are supporting a lot of coins.

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[LTU_btc]

OP, what you're planning to looks dangerous. Web wallet for long term storage isn't right place. Especially without 2FA. If you're really planning to keep your savings in web wallet, you should accept all possible risks.
People above already suggested you some web wallets. I know, it's your choice, but I think you should consider other options, like desktop wallet. It doesn't requires too much data to download and you'll have to do it only once. And you don't have to download whole blockchain and sync it, unless you're using Bitcoin Core wallet.

What I was actually thinking was that there are still people using very slow dial-up internet that have a maximum speed of 56 Kbps which is just 0.056 Mbps - and that would result in a download time of approximately 1 hours and 15 minutes for file size 30 MB.

Some may say, so who still uses dial-up internet access today, but research says that at least 1 million people in the USA use such internet today. It sounds strange for the 21st century, but for some it is quite enough, while others may not have a better choice.

People really still use dial-up internet? Actually, a million do

Wow, I didn't expected that in 2020 so many people still have such slow internet connection. I thought that even in least developped countries where they have problems with basic things like clean water, they have decent internet already.

[Lucius]

LTU_btc, I was also surprised with this information, but the fact is that such a service still exists and that at least for them it is relatively cheap ($10 per month) or even completely free 10 hours a day. However, for some $20 + I have about 40 Mbps with a landline phone that is unlimited to landline numbers and I still get 1000 minutes free to all mobile numbers within the country.

I believe that dial-up internet is used mostly by older people, and that the number will be smaller in the next few years, at least as far as the USA is concerned.