There is no downside of multiple cards. Take sapphiro ltf without thinking. Next you need one from hdfc & sbi each. What's your salary?
If your net salary >1Lpm in partner companies, you can get millenia easily. Outside partner companies they might ask for 1.5L. Know that with negotiation you can get with lower salary as well. If you dont succeed then you can take a lower card as ltf but the problem is Auto le and card upgrade is very hard in hdfc without good spend on their cards.
If you want just for offers and can't get any card as ltf, get a rupay card/fuel card. Easier to reach fee waiver.
Sadly sbi won't give any cc ltf. You may have to take a lower varinat card. Same rupay/fuel to reach fuel waiver. One thing you can do is all spend which doesn't give rewards in ace, can be transferred here. Fuel/govt/insurance etc. Sbi cb is best cb card but doesn't give discount during sales.
If you do too many hard enquiries, it will dip but it will recover in 2-3 months. So you plan to take homeland only after 3 months, then apply for rest. But take sapphiro right now as you Don't know whether you ll hey same offer later.