Lampard oversaw a squad overhaul at Stamford Bridge last summer, bringing in a host of new players.
The club smashed their transfer record by signing Kai Havertz in a deal which will be worth £90m, while Timo Werner and Chilwell both arrived for more than £50m.
Hakim Ziyech also signed in a deal worth £37.5m, as Chelsea looked to back Lampard after their transfer ban ended.
However, despite backing him financially, the club’s board gave him little wiggle room to get things going on the pitch.
Just four months after the transfer window closed, Lampard was fired by Chelsea after failing to close the gap on the top teams in the Premier League.
And Redknapp believes Lampard was only ever properly keen on Chilwell, despite the wealth of talent to come through the doors at Stamford Bridge.
Chilwell was ever-present during his time under Lampard, but was dropped for Thomas Tuchel’s second game in charge against Burnley on Sunday.
Writing for The Sun, Redknapp explained that transfer activity for managers has changed significantly since he was working and that Lampard would have been in a difficult position when it came to buying players.
“Years ago of course they would be the one scouting players, watching them play, basically doing the deals. Not now, though,” Redknapp wrote.
“These days it is pretty much all down to the head of recruitment, which means you are relying on someone else’s judgment.
“Chelsea spent fortunes on new players and as manager you’re under pressure — not only to play them — but for them also to be an instant success.
“When they don’t, like the two Germans Timo Werner and Kai Havertz, then the buck stops with the boss.
“But I’m sure Frank isn’t the one behind all the signings Chelsea have made.
“I’d say the only one he was mad keen on getting was Ben Chilwell. That’s how crazy the game has become.”